## 3 Results and Discussion
### Rapid Measurement System
#### 3.2.4 Comparison between CMM and Actual Rapid System Measurements  
The measurement results obtained from using the proposed system were compared with results obtained in an actual situation. To reduce the time taken to adjust the parameters when the machine is eventually moved to the factory, the factory provided CMM measurement data to use as the basis for machine adjustment/comparison.  
This study successfully developed a rapid optical measurement system that can reduce the inspection time from the 30 min of CMM to 60 s.  
Figure 12 shows the CMM measurement positions. Because the workpiece tooth groove is processed by a broach, the groove is tapered toward the top of its face. The CMM measurement is based on the middle position. In this study and in the projector measurement, the maximum size was measured because the optical measurement system can only measure the outermost shape projection. This section discusses the difference in dimensions obtained with the optical method and the CMM.  
Figure 6 shows the tooth positions, which are numbered. The key measurement point of the CMM is the maximum size of 16 roots (Rr) because this is the dimension where interference occurs during product assembly. As in Section 2.6.2, a simple direct filter was added to the data-processing  
Figure 11: Histogram showing thickness error distribution.  
Figure 12: Coordinate measuring machine (CMM) measurement positions and face taper.  
When the right correction value was used, the curve in Figure 13 is almost horizontal, and the oscillation is small; all measurement values were lower than those obtained with the CMM. The minimum error value is 0.005 mm, the error range is between 0.079 and 0.005 mm, and the total error value decreased to 0.075 mm.  
The measurement result indicates that the actual size of the workpiece is smaller than the measured size, and therefore, interference may not occur during actual assembly and work.  
The proposed optical measurement system can only measure the outermost shape projection of tapered face of the workpiece. Therefore, it is reasonable that the measurement results obtained from using our system will be bigger than the results obtained from using the CMM. Figure 13 shows the error between the correction results of two different circle centers and the CMM measurement result. When using the correct sample center, the error curve is smoother and approaches the horizontal, which indicates consistency in the trend of each error value; moreover, the total error is also small. When using an incorrect center correction, the curve oscillates like a sinusoid.  
The CMM measures the middle of the sample, and the dimensions measured by the system and the projector are the outermost contour dimensions of the workpiece. Therefore, differences in dimension and measured results occur. Results obtained with the CMM were compared with the results acquired from using optical projectors; the measurement errors are the same (\(-0.02\) mm) for the ring gauge and tooth shape samples. This means that our measurement system is accurate and stable. The result is satisfactory because the difference between the CMM measurement data and the results obtained with our system is 0.079 mm when using only the direct filter method.