# A New Tooth Profile Modification Method of Cycloidal Gears in Precision Reducers for Robots
## 5 Examples and Discussion  
The essence of the modification method in this paper is to obtain different tooth profile shapes by establishing different modification piecewise functions \(f(\alpha)\). In order to compare the different modification piecewise functions \(f(\alpha)\), the function curves obtained by the straight line method, cycloid method and catenary method are plotted in Figure 11. The geometric parameters of the cycloidal gear are shown in Table 1. According to the requirements of the cycloid-pin transmission, the predetermined tooth tip and root clearance are 0.02 mm and the modification value of the reference point is 0.005 mm. Through calculation, the minimum profile pressure angle is determined to be 42.24\({}^{\circ}\), which is also the pressure angle at the modification reference point. Since the change trend of the modification value from the reference point \(A\) to the tooth tip \(D\) is similar to that from the reference point \(A\) to the tooth root \(E\), only the function curve from the reference point \(A\) to the tooth tip \(D\) is plotted in Figure 11. Through substituting the pressure angle and modification value at the reference point and the tooth tip point into Equations (8) and (9), the modification curves are obtained (Figure 11). Figure 11: Modification curves of different methods. Figure 10: Process of solving the minimum backlash angle. The range of the working section of the cycloid-pin gear in the actual transmission process is relatively small. Similarly, the range of the profile pressure angle in the working section in Figure 11 is 42.24\({}^{\circ}\)\(\sim\)42.45\({}^{\circ}\). In this working section, the modifications calculated by the new method are not much different, and their changes are also slow. Therefore, it can be preliminarily judged that the modified profile near the reference point \(A\) is the closest to the theoretical profile in the meshing area, and the modification effect should be reasonable. According to Equations (3) and (4), the modifications are superimposed on the normal direction of the cycloidal profile, and the modified tooth profile of the cycloid gear and pin tooth profile in coordinate system \(S_{c}\) are obtained. With the aid of Equation (10), the meshing contact analysis of the cycloid-pin gear after the modification is completed and the transmission error curve and the minimum backlash angle curve are obtained (Figures 12 and 13). The traditional modification method adopts the modification combination of the positive equidistance and the positive displacement. The equidistant modification value is 0.005 mm and the displacement modification is 0.015 mm. These settings can ensure that the initial conditions of the traditional and new modification methods are the same for the convenience of comparison and discussion. In Figure 12, the curve segment \(B_{i}C_{i}\) (\(i\) = 1, 2, 3, 4) indicates an engagement period of the modified profile and Points \(B_{i}\) and \(C_{i}\), respectively, correspond to the meshing start and end points of the gear tooth. In the meshing area, the transmission error curves of the modified profiles are smooth and the symmetry is also good. The maximum transmission error of the traditional modification is \(-\)1.47 arcsec, which is significantly larger than that of the new method. The transmission error of the tooth profile modified by the second cycloid method is the smallest, which is \(-\)0.44 arcsec and indicates the transmission accuracy is higher. Figure 12: Transmission error curves of modified tooth profiles. Figure 13: Minimum backlash curves of modified tooth profiles. Under the same initial clearance requirements, the minimum backlash angle curve obtained by the new modification method is basically the same and the value of the minimum backlash angle has little difference, reaching approximately 0.6 arcmin (Figure 13). The minimum backlash angle of the traditional modification method is significantly larger, reaching 0.99 arcmin.