# A New Tooth Profile Modification Method of Cycloidal Gears in Precision Reducers for Robots
## 3 Calculation of the Modification of Tooth Profile  
### General Representation of the Modification  
The purpose of the tooth profile modification is to ensure that the modification in the meshing area (working segment) of cycloidal gear is as small as possible. The tooth profile in this working segment is as close as possible to the theoretical one, whereas the tooth profile in the non-working segment retains a reasonable backlash to meet lubrication and assembly requirements [3]. In this way, the profile modification strategy of a cycloidal gear can be established (Figure 4).  
In Figure 4, the \(BC\) segment of the tooth profile is the working segment and the \(BD\) and \(CE\) segments are the non-working segments. The modified curve in the \(BC\) segment is close to the theoretical conjugate tooth profile. From the modification reference point (Point \(A\)) to the tooth tip (Point \(D\)) or tooth root (Point \(E\)), the modification value gradually increases with the increase in the pressure angle. Therefore, Point \(A\) is considered as the reference point for profile modification and the modification value at Point \(A\) is also the smallest.  
The position selection of the reference point \(A\) is important in the modification design. According to the variations of the pressure angle on the theoretical tooth profile of a cycloid gear, the position of the reference point \(A\) is initially determined as the position with the smallest profile pressure angle to ensure that the reference point is in the meshing working segment of the tooth profile. Moreover, it ensures that the force transmission performance of the meshing segment is optimal. It should be noted that the modifications of reference point \(A\), the tooth tip point \(D\) and tooth root point \(E\) should be reasonably determined according to the backlash, assembly accuracy, processing accuracy, lubrication conditions and machine size of the cycloid-pin gear drive.  
According to the tooth profile modification strategy of cycloidal gear shown in Figure 4, the mathematical model for tooth profile modification can be established with pressure angle \(\alpha\) as independent variable, modification \(\Delta L\) as dependent variable, and the meshing clearances between  
Figure 4: Tooth profile modification strategy of a cycloidal gear.  
the reference point \(A\), the tooth tip point \(D\) and tooth root point \(E\) as the constraints. The functional relationship between the modification and the pressure angle can be expressed as follows:  
\[\left\{\begin{array}{l}\Delta L=f(\alpha)\\ s.t.\\ \Delta L_{tip}\leq C_{tip}\\ \Delta L_{root}\leq C_{root}\\ \Delta L_{0}\leq C_{0}\end{array}\right., \tag{6}\]  
where \(\Delta L=f(\alpha)\) is the objective function of tooth profile modification optimization, a piecewise function related to the distribution of the pressure angle; \(C_{0}\), \(C_{tip}\). and \(C_{root}\) are, respectively, the clearance index of reference point \(A\), the tip point \(D\) and root point \(E\); \(\Delta L_{0}\), \(\Delta L_{tip}\) and \(\Delta L_{root}\) are, respectively, the modification values of reference point \(A\), the tip point \(D\) and root point \(E\).  
Different methods of determining the piecewise function \(f(\alpha)\) correspond to different mathematical relations of modifications. The ideal profile curve and transmission accuracy can be obtained by the modification function \(f(\alpha)\) determined with mathematical curves such as straight line, cycloid line and catenary line. In this paper, the modification calculation of the cycloid method was adopted as an example.