## 2 Error Tooth Surface Model of Cycloid Gear and Pin
The different AEs can be obtained using different combinations of the above four parameters. The coordinates of the element control point \(i_{\text{c}}\times j_{\text{c}}\) on the \(n_{\text{c}}^{\text{th}}\) tooth surface of the cycloid gear considering AEs are given as follows:  
\[\left[ \begin{array}{cccc}X_{\text{c}}^{n_{\text{c}}i_{\text{c}}j_{\text{c} }}&Y_{\text{c}}^{n_{\text{c}}i_{\text{c}}j_{\text{c}}}&Z_{\text{c}}^{n_{\text {c}}i_{\text{c}}j_{\text{c}}}&1\end{array} \right]^{\text{T}}=\mathbf{M}_{ \text{ae}}\Big{[} \begin{array}{cccc}X_{\text{c}}^{n_{\text{c}}i_{\text{c}}j_ {\text{c}}}&Y_{\text{c}}^{n_{\text{c}}i_{\text{c}}j_{\text{c}}}&Z_{\text{c} 2}^{n_{\text{c}}i_{\text{c}}j_{\text{c}}}&1\end{array} \Big{]}^{\text{T}}, \tag{6}\]  
where \(\mathbf{M}_{\text{ae}}\) is the coordinate transformation matrix considering AEs, which is given as follows:  
\[\mathbf{M}_{\text{ae}}=\left[ \begin{array}{cccc}1&0&(1-\sin\gamma)\tan\varphi -\sin\varphi&\Delta x+B_{\text{c}}[\sin\varphi-(1-\sin\gamma)\tan\varphi]\\ 0&1&-\sin\gamma/\cos\varphi&\Delta y+B_{\text{c}}\sin\gamma/\cos\varphi\\ 0&0&\sin\gamma-\cos\varphi-1&B_{\text{c}}(\cos\varphi+1-\sin\gamma)\\ 0&0&0&1\end{array} \right]. \tag{7}\]  
To summarize, the coordinates of each element control point of the cycloid gear and pin considering PMs, MEs, and AEs have been obtained. Additionally, the radius of curvature and the normal vector at each control point, as well as the axis direction vector of the approximate cylinder corresponding to each element are also necessary to uniquely determine the position and size of the equivalent cylindrical surface corresponding to each element. The radii of curvature at each element control point of the cycloid gear and pin are given as follows:  
Figure 3: (**a**) Part of the tooth surface of cycloid gear considering PMs and MEs. (**b**) Definition of AEs. \[\rho_{\rm c}^{n_{\rm c}i_{\rm c}j_{\rm c}}=\frac{\left[e^{2}z_{\rm p}^{2}+\left(R_{ \rm p}+\Delta R_{\rm p}\right)^{2}-2 ez_{\rm p}\left(R_{\rm p}+\Delta R_{\rm p} \right)\cos\left(\phi_{\rm P}^{n_{\rm c}i_{\rm c}j_{\rm c}}\right)\right]^{3/2} }{e^{2}z_{\rm p}^{3}+\left(R_{\rm p}+\Delta R_{\rm p}\right)^{2}-ez_{\rm p}\left(1+ z_{\rm p}\right)\left(R_{\rm p}+\Delta R_{\rm p}\right)\cos\left(\phi_{\rm P}^{n_{ \rm c}i_{\rm c}j_{\rm c}}\right)}-\left(R_{\rm rp}+\Delta R_{\rm rp}+\delta^{ n_{\rm c}i_{\rm c}j_{\rm c}}\right), \tag{8}\]  
\[\rho_{\rm p}^{n_{\rm p}i_{\rm p}j_{\rm p}}=R_{\rm rp}+\delta^{n_{\rm p}i_{\rm p }j_{\rm p}}, \tag{9}\]  
where \(\rho_{\rm c}^{n_{\rm c}i_{\rm c}j_{\rm c}}\) can be negative (\(-\)) for a concave surface and positive (\(+\)) for a convex surface. The normal vectors at each element control point of the cycloid gear and pin are respectively given as follows:  
\[\mathbf{n}_{\rm c}^{n_{\rm c}i_{\rm c}j_{\rm c}}=\mathbf{M}_{\rm ae}\Big{[} \begin{array}{cc}dX_{\rm c1}^{n_{\rm c}i_{\rm c}j_{\rm c}}/d\phi_{\rm P}^{n_ {\rm c}i_{\rm c}j_{\rm c}}&dY_{\rm c1}^{n_{\rm c}i_{\rm c}j_{\rm c}}/d\phi_{\rm P }^{n_{\rm c}i_{\rm c}j_{\rm c}}&0&1\end{array} \Big{]}^{\rm T}, \tag{10}\]  
\[\mathbf{n}_{\rm p}^{n_{\rm p}i_{\rm p}j_{\rm p}}=\big{[} \begin{array}{cc} \cos\left(2\pi\left(j_{\rm p}-1/2\right)/n_{2\rm p}\right)&\sin\left(2\pi\left( j_{\rm p}-1/2\right)/n_{2\rm p}\right)&0&1\end{array} \big{]}^{\rm T}. \tag{11}\]  
The axis direction vectors of the approximate cylinder of each element of the cycloid gear and pin are given as follows:  
\[\mathbf{V}_{\rm c}^{n_{\rm c}i_{\rm c}j_{\rm c}}=\big{[} \begin{array}{cc} \left(1-\sin\gamma\right)\tan\varphi-\sin\varphi&-\sin\gamma/\cos\varphi&\sin \gamma-\cos\varphi&1\end{array} \big{]}^{\rm T}, \tag{12}\]  
\[\mathbf{V}_{\rm P}^{n_{\rm p}i_{\rm p}j_{\rm p}}=\big{[} \begin{array}{cccc} 0&0&1&1\end{array} \big{]}^{\rm T}. \tag{13}\]