# A New Tooth Profile Modification Method of Cycloidal Gears in Precision Reducers for Robots
## 2 Theoretical Tooth Profile and Pressure Angle of a Cycloid Gear
### Pressure Angle Distribution of Cycloid Profile  
The pressure angle of tooth profile refers to the intersegment angle between the force direction at the meshing point and the velocity direction of this point without considering the friction force [7]. It is also the angle between the common normal direction at the meshing point and the rotation velocity direction of the cycloidal gear, as shown in Figure 2.  
In Figure 2, Point \(K\) is a meshing contact point in coordinate system \(S_{c}\) and the pressure angle \(\alpha\) of the point \(K\) is the angle between the normal direction \(\mathbf{n}_{c}\) and the velocity direction \(\mathbf{v}_{c}\). The pressure angle can be expressed as:  
\[\alpha=arccos\big{|}\big{(}\mathbf{v}_{c}\cdot\mathbf{n}_{c}\big{)}\big{|}= arccos\left|\left(\frac{\overset{\rightarrow}{O_{c}K\times\mathbf{k}}}{ \big{|}\overset{\rightarrow}{O_{c}K\times\mathbf{k}}}\cdot\mathbf{n}_{c} \right)\right|, \tag{5}\]  
where \(\mathbf{k}=\left[\begin{array}{cc}0&0&1\end{array}\right]^{T}\) is perpendicular to the plane \(X_{c}Y_{c}\).  
The pressure angle of each point on the theoretical tooth profile of the cycloidal gear can be calculated with Equation (5). Then the distribution trend of the pressure angle on the tooth profile can be obtained. Figure 3 shows the pressure angle distribution of a single tooth profile of the cycloidal gear. The related geometric parameters of the cycloid-pin gear are shown in Table 1.  
Obviously, the pressure angle in Figure 3 generally shows a \"high-low-high\" distribution trend. The pressure angle is the maximum at the tip and root of teeth respectively, and reaches the minimum at Point \(A\). From the definition of the pressure angle and the force analysis, the larger pressure angle of tooth profile indicates the larger angle between the common normal direction and the velocity direction at the meshing point and the poorer force transmission performance of the gear teeth. The  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Symbols** & **Values** \\ \hline \hline \begin{tabular}{c} Tooth number of cycloidal gear \\ \end{tabular} & \(z_{c}\) & 39 \\ \begin{tabular}{c} Tooth number of pin gear \\ \end{tabular} & \(z_{p}\) & 40 \\ \begin{tabular}{c} Roller radius \\ \end{tabular} & \(r_{rp}\) & 3.5 mm \\ \
\begin{tabular}{c} Eccentricity \\ \end{tabular} & \(a\) & 1.5 mm \\
\begin{tabular}{c} Roller position \\ \end{tabular} & \(r_{p}\) & 82 mm \\ \hline \hline \end{tabular}
\end{table}
Table 1: Geometric parameters of the cycloid-pin gear.  
Figure 3: Pressure angle distribution of the single tooth profile.  
Figure 2: Pressure angle of the cycloidal profile.  
smaller pressure angle indicates the better transmission performance. It can be seen from Figure 3 that when the pin gear rotation angle is \(137.49^{\circ}\), the pressure angle reaches the minimum value, \(42.24^{\circ}\). Therefore, the transmission performance is optimal at this position. Point \(A\) is used as a reference point in the tooth profile modification design.