# 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  
### Expression of the Theoretical Tooth Profile  
In the cycloid-pin gear planetary transmission of precision reducers for robots, the cycloidal gear rotates around its own axis as well as the rotation axis of the pin gear, thus generating the conjugate meshing motion between the tooth profiles of the cycloid-pin gear. Therefore, the meshing motion relationship of a cycloid-pin gear can be established, as shown in Figure 1. Coordinate system \(S_{f}\big{(}x_{f},y_{f}\big{)}\) is introduced and fixed on the frame. Coordinate systems \(S_{p}\big{(}x_{p},y_{p}\big{)}\) and \(S_{c}(x_{c},y_{c})\) are respectively defined for the pin gear and the cycloidal gear. The pin gear acts as the driving wheel and its axis rotates around the center \(O_{p}\). The cycloidal gear acts as the driven wheel and its axis rotates around the center \(O_{c}\). The rotation center \(O_{p}\) of the pin gear coincides with the origin \(O_{f}\) of coordinate system \(S_{f}\). \(a\) is the eccentricity of cycloidal gear and pin gear. \(r_{p}\) and \(r_{rp}\) represent the roller radius and roller position, respectively. Point \(K\) is the current meshing position and Point \(P\) is the instantaneous center. Line \(PM\) is the common normal of the meshing point \(K\). Angles \(\phi_{1}\) and \(\phi_{2}\) are the rotation angles of the pin gear and the cycloid gear, respectively.  
In coordinate system \(S_{p}\), without considering the tooth width, the pin tooth profile is a circle, with Point \(M\) as the center and \(r_{rp}\) as the radius. The tooth profile equation of the pin gear in \(S_{p}\) can be expressed as:  
\[\textbf{R}_{p}(\beta)=\Big{[}\begin{array}{ccc}-r_{rp}\sin\beta&r_{p}-r_{rp }\cos\beta&0&1\end{array}\Big{]}^{T}, \tag{1}\]  
where \(\beta\) is the parametric angle of the pin tooth. In order to facilitate the analysis and calculation of gear tooth contact, Equation (1) is expressed in the form of homogeneous coordinates and the following equations are also expressed in the same form.  
Figure 1: Meshing motion relationship of cycloidal-pin gear.  
According to the conjugate meshing relationship between tooth profiles of cycloid-pin gear, the theoretical tooth profile equation of cycloidal gear in coordinate system \(S_{c}\) can be obtained after the coordinate transformation of the pin gear tooth profiles:  
\[\mathbf{R}_{c0}(\varphi)=\left[\begin{array}{c}\left(r_{p}-r_{ rp}/S\right)\sin\varphi-\left(a-k_{1}r_{rp}/S\right)\sin\left(z_{p}\varphi\right)\\ \left(r_{p}-r_{rp}/S\right)\cos\varphi-\left(a-k_{1}r_{rp}/S\right)\cos\left(z_ {p}\varphi\right)\\ 0\\ 1\end{array}\right], \tag{2}\]  
where \(\varphi\) is the parametric angle of the tooth profile of the cycloidal gear; \(k_{1}\) is called short width coefficient, \(k_{1}=az_{p}/r_{p}\); \(S=\sqrt{1+k_{1}^{2}-2k_{1}\cos(z_{c}\varphi)}\).  
When the cycloidal gear and the pin gear engage in the conjugate mode, the direction of the common normal \(PM\) at the contact point \(K\) is the force direction of the gear teeth meshing, as shown in Figure 1. The extension line of the common normal \(PM\) can be drawn and intersected with the \(Y_{c}\) axis at Point \(M^{\prime}\). In \(\Delta M^{\prime}OM\), \(\Delta M^{\prime}OM=\phi_{2}-\phi_{1}\). The direction cosine of the common normal \(PM\) in coordinate system \(S_{c}\) can be obtained as:  
\[\mathbf{n}_{c}(\varphi)=\left[\begin{array}{c}\left(k_{1}\sin\left(z_{p} \varphi\right)-\sin\varphi\right)/\sqrt{1+k_{1}^{2}-2k_{1}\cos(z_{c}\varphi)}\\ \left(-k_{1}\cos\left(z_{p}\varphi\right)+\cos\varphi\right)/\sqrt{1+k_{1}^{2} -2k_{1}\cos(z_{c}\varphi)}\\ 0\\ 1\end{array}\right]. \tag{3}\]