## 2 Design of Perfect Surface  
The basic principle for the design of the perfect surface is as follows. Firstly, the pinion auxiliary tooth surface is built based on the preset TE. Secondly, the CP on pinion surface and its modification curve along the instantaneous meshing line are preset based on the design requirements. Thirdly, the correction value of the auxiliary pinion surface along the instantaneous meshing line is calculated, and it is superimposed on the pinion auxiliary surface. After this, the perfect pinion surface, which satisfied the functional requirements, is obtained. The detailed process is as follows.  
According to the designed contact ratio, the CP is designed as follows:  
\[y=(x-x_{M})\tan\eta+y_{M} \tag{1}\]  
where \(x_{M}\) and \(y_{M}\) denote the coordinate of reference point, and \(\eta\) is the direction angle.  
According to the functional requirements, the transmission error curve of the SBG transmission is designed as follows:  
\[\delta\varphi_{2}=0.5m^{\prime}{}_{21}(\varphi_{1}-\varphi_{1}^{(0)})^{2} \tag{2}\]  
where \(\varphi_{1}\) and \(\varphi_{1}\)(\({}^{0}\)) denote the actual and original pinion meshing angle, and \(m^{\prime}{}_{21}\) denotes the first derivative of the contact ratio function.  
According to the machining theory, the gear surface is built as follows:  
\[\left\{\begin{array}{l}\mathbf{r}_{2}=\mathbf{r}_{2}(s_{g},\theta_{g},\varphi_{g}) \\ \mathbf{n}_{2}=\mathbf{n}_{2}(s_{g},\theta_{g},\varphi_{g})\\ f_{2}(s_{g},\theta_{g},\varphi_{g})=0\end{array}\right. \tag{3}\]  
where \(\mathbf{n}_{2}\) and \(\mathbf{r}_{2}\) denote gear normal and position vector, \(\theta_{g}\) and \(s_{g}\) are the flank parameter, \(f_{2}\) denotes the meshing equation, and \(\varphi_{g}\) is the gear cradle angle.  
According to the meshing process of the SBG transmission, the auxiliary pinion surface is generated with the gear surface used as the cutter as follows:  
\[\left\{\begin{array}{l}\mathbf{r^{\prime}}_{1}=\mathbf{M}_{1h}(\varphi_{1})\mathbf{M}_{ h2}(\varphi_{2})\mathbf{r}_{2}\\ \mathbf{n^{\prime}}_{1}=\mathbf{L}_{1h}(\varphi_{1})\mathbf{L}_{h2}(\varphi_{2})\mathbf{n}_{2} \\ f_{12}(s_{g},\theta_{g},\varphi_{g},\varphi_{1})=0\end{array}\right. \tag{4}\]  
where \(\varphi_{1}\) and \(\varphi_{2}\), respectively, denote the pinion meshing angle and the gear cutter meshing angle, and \(f_{12}\) denotes the meshing equation. \(\mathbf{M}_{h2}\) and \(\mathbf{M}_{1h}\) are the coordinate transform matrices. \(\mathbf{L}_{1h}\) and \(\mathbf{L}_{h2}\) are the upper-left 3 \(\times\) 3 submatrices of \(\mathbf{M}_{1h}\) and \(\mathbf{M}_{h2}\).  
\[M_{h2}=\begin{bmatrix}\cos\Sigma&\sin\Sigma\sin\varphi_{2}&-\sin\Sigma\cos\varphi_{2}& 0\\ 0&-\cos\varphi_{2}&-\sin\varphi_{2}&0\\ -\sin\Sigma&\cos\Sigma\sin\varphi_{2}&-\cos\Sigma\cos\varphi_{2}&0\\ 0&0&0&1\end{bmatrix}\mathbf{M}_{1h}=\begin{bmatrix}1&0&0&0\\ 0&\cos\varphi_{1}&\sin\varphi_{1}&0\\ 0&-\sin\varphi_{1}&\cos\varphi_{1}&0\\ 0&0&0&1\end{bmatrix}\]  
To obtain the perfect pinion surface shown in Figure 1, the auxiliary pinion surface obtained through Equation (4) needs to be modified along the meshing line. And in Figure 1, the blue line is the perfect pinion surface, and the black line is the auxiliary pinion surface.  
The position vectors of the perfect surface (Figure 1) are obtained as follows:  
\[\mathbf{p}_{i}^{*}=\mathbf{p^{\prime}}_{i}+\mathbf{n^{\prime}}_{i}\delta_{yi} \tag{5}\]  
where \(\mathbf{p^{\prime}}\) and \(\mathbf{n^{\prime}}\) denote the position and normal vectors of the auxiliary pinion surface, and \(\delta_{y}\) denotes the correction.