## 3 Load Distribution Model
### Unloaded Tooth Contact Analysis  
The unloaded tooth contact analysis (TCA) is developed in the two-dimensional plane in order to instantly determine the instantaneous central points of potential contact regions. Figure 6 shows the coordinate systems for unloaded TCA. \(S_{1}(x_{1},y_{1})\) and \(S_{2}(x_{2},y_{2})\) are connected rigidly with the pinwheel and cycloid gear, and the fixed coordinate system \(S_{f}(x_{f},y_{f})\) coinciding with \(S_{1}\) is fixed to the housing. Hence, the position vector \(\mathbf{r}_{f}^{(1)}(\theta_{ri})\) and its unit normal vector \(\mathbf{n}_{f}^{(1)}(\theta_{ri})\) of all the roller profiles can be expressed in \(S_{f}\):  
\[\mathbf{r}_{f}^{(1)}(\theta_{ri})=\left[\begin{array}{c}a\cos(i\psi_{r})+ \rho\cos(\theta_{ri})\\ a\sin(i\psi_{r})-\rho\sin(\theta_{ri})\\ 0\end{array}\right], \tag{8}\]  
\[\mathbf{n}_{f}^{(1)}(\theta_{ri})=\frac{\frac{\mathbf{r}_{f}^{(1)}(\theta_{ri })}{\phi\theta_{ri}}\times\mathbf{k}}{\left|\frac{\mathbf{r}_{f}^{(1)}(\theta_ {ri})}{\phi\theta_{ri}}\times\mathbf{k}\right|}, \tag{9}\]  
where \(i\) is the number of rollers, \(\mathbf{k}=[0,0,1]\) is a unit vector, and \(\theta_{ri}\) is the angle parameter of the roller profile.  
Similarly, the position vector \(\mathbf{r}_{f}^{(2)}(\theta_{ci},\phi_{ci},\phi_{in})\) and its unit normal vector \(\mathbf{n}_{f}^{(2)}(\theta_{ci},\phi_{ci})\) can be represented in \(S_{f}\):  
\[\mathbf{r}_{f}^{(2)}(\theta_{ci},\phi_{ci},\phi_{in})=\left[\begin{array}{c} (a+\Delta a)\cos(\phi_{ci}-\theta_{ci})+(\rho+\Delta\rho)\cos(\theta_{m}+\phi _{ci}-\theta_{ci})+e[\cos\phi_{in}-\cos(\phi_{ci}-n_{1}\theta_{ci})]\\ -(a+\Delta a)\sin(\phi_{ci}-\theta_{ci})-(\rho+\Delta\rho)\sin(\theta_{m}+\phi _{ci}-\theta_{ci})+e[\sin\phi_{in}+\sin(\phi_{ci}-n_{1}\theta_{ci})]\\ 0\end{array}\right], \tag{10}\]  
\[\mathbf{n}_{f}^{(2)}(\theta_{ci})=\frac{\frac{\mathbf{r}_{f}^{(2)}(\theta_{ ci})}{\phi\theta_{ci}}\times\mathbf{k}}{\left|\frac{\mathbf{r}_{f}^{(2)}( \theta_{ci})}{\phi\theta_{ci}}\times\mathbf{k}\right|}, \tag{11}\]  
where \(\phi_{ci}\) is the rotational angle of the cycloid gear tooth, \(\phi_{in}\) is the input angle of the crankshaft, and \(\theta_{ci}\) is the angle parameter of the cycloid tooth profile.  
Figure 5: A flowchart of the load distribution analysis methodology for cycloid drives.  
To perform the unloaded TCA, two conditions must be satisfied, which are coincidence of the position vectors and collinearity of the normal vectors of the modified profile of the cycloid gear and roller profiles at the same contact points [15; 25]. Thus,  
\[\begin{array}{l}\mathbf{n}_{f}^{(1)}(\theta_{ri})=\mathbf{n}_{f}^{(2)}(\theta _{ci},\phi_{ci})\\ \mathbf{r}_{f}^{(1)}(\theta_{ri})=\mathbf{r}_{f}^{(2)}(\theta_{ci},\phi_{ci}, \phi_{in})\end{array} \tag{12}\]  
Since for the unit normal vectors, \(\left|\mathbf{n}_{f}^{(1)}\right|=\left|\mathbf{n}_{f}^{(2)}\right|=1\), three nonlinear equations with the four unknowns \(\theta_{ri}\), \(\theta_{ci}\), \(\phi_{ci}\), and \(\phi_{in}\) can be developed. Given one, the input crankshaft angle \(\phi_{in}\), the other three unknown parameters can be solved.  
Then, the backlash \(\phi_{bli}\) and the rotational angle \(\phi_{c}\) of the cycloid gear can be expressed as follows [5; 15]:  
\[\begin{array}{l}\phi_{bli}=\phi_{c}-\phi_{ci}\\ \phi_{c}=\text{max}(\phi_{ci})\end{array} \tag{13}\]  
where the backlash \(\phi_{bli}\) is defined as the required rotational angle for all the potential meshing tooth pairs, and the rotational angle \(\phi_{c}\) of the cycloid gear is the maximum value among the rotational angles of all the cycloid gear teeth.