# On-Machine Precision Form Truing and In-Situ Measurement of Resin-Bonded Spherical Diamond Wheel
## 3 Measurement and Compensation of the Alignment Error  
The position of the spherical diamond wheel and the laser scan micrometer in the measuring process is shown in Figure 5a. The laser beam by the laser scan micrometer is adjusted to parallel to the _yoz_-plane of the machine tool coordinate system, and coplanar with the axis of the diamond wheel. Then, the laser scan micrometer is controlled by the CNC system to scan the diamond wheel surface bottom-up linearly. The scanning direction is perpendicular to the axis of the diamond wheel. The sampling points of the laser scan micrometer are the height information of the diamond wheel surface, and the continuous sampling data constitute the profile of the spherical diamond wheel. As shown in Figure 5b, the arc AC in the measurement results corresponds to the AC section profile of the spherical diamond wheel. After truing the spherical diamond wheel, the AC section is tested, and the least square fitting is performed. The fitting radius and residual value are the measuring radius and profile error of the spherical diamond wheel, respectively.  
Based on the measurement results shown in Figure 5b, a biarc curve-fitting method is proposed to calculate the alignment error between the truing spindle and tool spindle. As displayed in Figure 6, the arc A'B'C' represents the ideal spherical diamond wheel profile, and the arc ABC is the actual diamond roller trued profile. The DE arc is intercepted and fitted from the AB segment of the detected profile, and the coordinates \((x_{l},y_{l})\) of the circle center \(O_{l}\) as well as the arc radius \(r_{l}\), are then calculated. Similarly, the GF arc is intercepted and fitted from the BC segment of the detected profile, and the position coordinates \((x_{r},\,y_{r})\) of the circle center \(O_{r}\) as well as the arc radius \(r_{r}\) are then obtained. The distance \(O_{l}O_{r}\) between the center of the two circles is the alignment error in the \(x\) direction. When \(x_{r}\)\(>\)\(x_{l}\), the roller is not trued to the center of the spherical diamond wheel, as shown in Figure 6a. When \(x_{r}\)\(<\)\(x_{l}\), the roller has passed by the center of the spherical diamond wheel, as shown in Figure 6b.  
Figure 5: In-situ measurement of spherical diamond wheel profile: (**a**) measurement, (**b**) results.  
Figure 6: Schematic of the biarc curve-fitting method: (**a**) diamond roller is before the center, (**b**) diamond roller is past the center.  
Figure 7 shows a set of center coordinates and radius fitting values obtained by the biarc curve-fitting method. It is seen that the alignment error between the roller and the spherical diamond wheel is approximately 0.28 mm, and the diamond roller is not trued to the center of the spherical diamond wheel. The alignment error can be eliminated by introducing a \"truing-detection-compensation\" process. Then, the precision form truing of the resin-bonded spherical diamond wheel can be achieved. The diameter of diamond roller can be approximately considered as fixed in the actual truing process of the resin-bonded diamond wheel.  
Figure 8 displays scanning electron microscope (SEM) micrographs showing the surface topography of the resin-bonded spherical diamond wheel after the diamond roller truing. The flat and regular surface of the spherical diamond wheel proves the good truing effect of the diamond roller. Part of the diamond grains were broken during the truing process, and the micro-breakage of the abrasive grains generates new sharp cutting edges. The diamond roller truing method can achieve the purposes of both shaping and sharpening. It is conducive to enhancing the grinding ability of the spherical diamond wheel, and improving both the ground surface form accuracy and the surface quality.