## 2 Grinding Surface Topography of the Screw Raceway
### Modeling of the Geometric Model of the Screw Raceway
In the coordinate system \(AX_{A}Y_{A}Z_{A}\), the \(Z_{A}\)-axis is perpendicular to the tangent plane of the contact point \(A\), the \(X_{A}\)-axis is parallel to the tangent direction of the spiral line of the ball center, and the \(Y_{A}\)-axis direction is determined by the right-hand rule. The parameter equation of the ball surface topography in the contact coordinate system \(AX_{A}Y_{A}Z_{A}\) is expressed as follows:  
\[z_{k}=r_{b}-\sqrt{{r_{b}}^{2}-{x_{k}}^{2}-{y_{k}}^{2}},k=A \tag{15}\]  
where \(A\) is the contact point between the ball and the screw. The parameter equation of the geometric morphology of the raceway surface is obtained according to the coordinate transformation method in the contact coordinate system \(AX_{A}Y_{A}Z_{A}\). In the Frenet-Serret coordinate system \(O_{H}tnb\), the parameter expression of the cross-sectional arc of the screw raceway at contact point \(A\) is \(\textbf{P}_{AS}^{H}\). The expression of the transformation matrixes is given in Appendix A. The parameter equation of the screw raceway in the coordinate system \(OXYZ\) is expressed as \(\textbf{P}_{AS}^{\textbf{S}}=\textbf{T}_{S}^{H}\textbf{P}_{AS}^{H}\), where \(\textbf{T}_{S}^{H}\) denotes the coordinate transformation matrix from the coordinate system \(OXYZ\) to the Frenet-Serret coordinate system \(O_{H}tnb\) of the helical trajectory of the ball center. The coordinate transformation matrix from the Frenet-Serret coordinate system OHtnb to the contact coordinate system \(AX_{A}Y_{A}Z_{A}\) is \(\textbf{T}_{\textbf{H}}^{A}\). The parameter equation of the screw raceway in the \(AX_{A}Y_{A}Z_{A}\) contact coordinate system is established according to the coordinate transformation theory as follows:  
\[\textbf{P}_{AS}^{A}=\left(\textbf{T}_{S}^{H}(\theta_{C})\textbf{T}_{H}^{A} \right)^{-1}\textbf{P}_{AS}^{\textbf{S}},\beta_{A}-\beta_{A0}\leq\beta_{AS} \leq\beta_{A}+\beta_{A0} \tag{16}\]  
where \(\textbf{T}_{S}^{H}(\theta_{C})\) represents the coordinate transformation matrix when \(\theta=\theta_{C}\). \(\theta_{C}\) is the central angle corresponding to the spiral line at the contact point \(A\) between the ball and the screw raceway. The geometric morphology of the screw raceway is determined according to the parameter equation of the screw raceway. Substituting the values of parameters \(\theta\) and \(\beta_{A0}\) into the parameter equation of the screw raceway in the \(AX_{A}Y_{A}Z_{A}\) contact coordinate system, the geometric morphology of the screw raceway in the contact area is calculated. The structural and grinding parameters of the ball screws are determined according to the product manual [35] and the most common grinding process of the ball screws, as shown in \[d_{s,e}=d_{s},d_{w,e}=d_{w}/\cos^{2}\alpha \tag{12}\]0  
The projection of the three-dimensional geometric morphology of the screw raceway on the plane \(AX_{A}Y_{A}\) of the contact coordinate system is an irregular rectangular region. Therefore, the two-dimensional interpolation algorithm is used to re-divide the grid, and the interpolation result of the geometric morphology of the screw raceway is obtained, as shown in Figure 6a. The grinding surface topography of the screw raceway is simulated based on the formation mechanism of the grinding surface and the geometric morphology model of the screw raceway. The grinding surface topography of the screw raceway is shown in Figure 6b,c. \[d_{s,e}=d_{s},d_{w,e}=d_{w}/\cos^{2}\alpha \tag{12}\]1 Contact coordinate system of the ball screws. \[d_{s,e}=d_{s},d_{w,e}=d_{w}/\cos^{2}\alpha \tag{12}\]2 The simulated surface topography of the screw raceway: (**a**) Interpolation results of the geometric morphology of the screw raceway; (**b**) Surface topography of the grinding raceway; (**c**) Local enlargement of the surface topography of the grinding raceway; (**d**) The simulated surface topography profile with the radius removal.