## 2 TE Model of the RV Speed Reducer
### Geometric TE Analysis Model of the Cycloidal Gear Transmission  
The initial clearance is an internal factor of the geometric TE. Main factors of component errors that affect the geometric TE of the cycloidal gear transmission are as follows:  
1. Isometric and shifting tooth profile modifications of the cycloidal gear;
2. TE caused by a radius error of the center circle of pinwheels;
3. TE caused by an eccentricity error;
4. TE caused by a radial runout error of the cycloidal gear;
5. TE caused by a radius error of a pinwheel and the clearance between a pinwheel and its groove;
6. Clearance caused by a cumulative error of the circumference of the cycloidal gear and circumferential position errors of pinwheel housing;
7. Clearance caused by a modification error of the cycloidal gear; and
8. Rotor bearing clearance.  
Since a parallelism error between axes of the eccentric shaft and rotating shaft only causes a small clearance, it is neglected in this study. TEs caused by the above main factors of component errors that affect the geometric TE of a cycloidal gear transmission are illustrated as follows:  
A. The geometric TE caused by isometric tooth profile modification \(\Delta r_{rp}\) of the cycloidal gear can be represented as  
\[\Delta\theta_{a}=\frac{2\Delta r_{rp}}{az_{c}}. \tag{5}\]  
Figure 2: Coordinate systems of the cycloidal gear and pinwheel.  
B. The geometric TE caused by shifting tooth profile modification \(\Delta r_{p}\) of the cycloidal gear can be represented as  
\[\Delta\theta_{b}=-\frac{2\Delta r_{p}}{az_{c}}\sqrt{1-{K^{\prime}_{1}}^{2}}. \tag{6}\]  
C. The geometric TE caused by the radius error \(\delta_{r_{p}}\) of the center circle of pinwheels can be represented as  
\[\Delta\theta_{c}=\frac{2\delta_{r_{p}}}{az_{c}}\sqrt{1-{K^{\prime}_{1}}^{2}}. \tag{7}\]  
D. The geometric TE caused by the pinwheel radius error \(\delta_{r_{rp}}\) can be represented as  
\[\Delta\theta_{d}=-\frac{2}{az_{c}}\delta_{r_{p}}. \tag{8}\]  
E. The geometric TE caused by the matching clearance \(\delta_{j}\) between the pinwheel and pinwheel groove can be represented as  
\[\Delta\theta_{e}=\frac{\delta_{j}}{az_{c}}. \tag{9}\]  
F. The geometric TE caused by the radial runout error \(F_{r_{1}}\) of the cycloidal gear ring can be represented as  
\[\Delta\theta_{f}=\frac{F_{r_{1}}}{2 az_{c}}. \tag{10}\]  
G. The geometric TE caused by the circumferential position error \(\delta_{t_{\Sigma}}\) of a pinwheel housing can be represented as  
\[\Delta\theta_{g}=\frac{2K_{1}\delta_{t_{\Sigma}}}{az_{c}}. \tag{11}\]  
H. The geometric TE caused by the cumulative error \(\Delta F_{p}\) of the circumference of the cycloidal gear can be represented as  
\[\Delta\theta_{h}=-\frac{K^{\prime}_{1}\Delta F_{p}}{az_{c}}. \tag{12}\]  
I. The geometric TE caused by tooth profile modification errors, i.e., \(\delta\Delta r_{rp}\) and \(\delta\Delta r_{p}\), and the eccentricity error \(\delta_{\alpha}\) can be represented as  
\[\Delta\theta_{i}=\frac{2\delta\Delta r_{rp}}{az_{c}}-\frac{2\delta\Delta r_{p }}{az_{c}}\sqrt{1-{K^{\prime}_{1}}^{2}}-2k_{n}\delta_{\alpha}, \tag{13}\]  
where  
\[k_{n}=\frac{\Delta r_{rp}}{a^{2}z_{c}}-(\frac{z_{c}\Delta r_{p}}{{a_{p}}^{2} \sqrt{1-{K^{\prime}_{1}}^{2}}}+\frac{\Delta r_{p}}{a^{2}z_{c}}\sqrt{1-{K^{ \prime}_{1}}^{2}}). \tag{14}\]  
In summary, the geometric TE of the cycloidal gear transmission caused by the above factors can be represented as  
\[\Delta\theta=\Delta\overline{\theta}\pm\frac{T_{\Delta\theta}}{2}, \tag{15}\]  
where \(\Delta\overline{\theta}\) is the mean value of the geometric TE of the cycloidal gear transmission that can be represented as  
\[\Delta\overline{\theta}=\frac{180\times 60}{\pi}{\sum_{j=1}^{8}\Delta\theta_{j}}, \tag{16}\]and \(T_{\Delta\theta}\) is the tolerance of the geometric TE of the cycloidal gear that can be represented as  
\[T_{\Delta\theta}=\frac{180\times 60}{\pi}\sqrt{\sum_{j=1}^{8}\left(T_{\Delta \theta_{j}}\right)^{2}}, \tag{17}\]  
in which \(T_{\Delta\theta_{j}}\) is the tolerance of each error factor.