## 4 Conclusions  
We successfully developed a rapid optical measurement system that can reduce inspection time from 30 min to 30 s for 360\({}^{\circ}\) measuring time (total 60 s including output the result). This system can quickly identify the cutting tools that need adjustment or replacement, to maintain product dimension accuracy and yield.  
Measurements were first conducted by using a prototype system in the laboratory. The rotation speed was slow, and the measurement time was long. Moreover, the amount of data was large, and the resulting tooth thickness measurement error was small. The signal stability of the system was good; no glitches were observed. A wavelet periodic signal with 80 noise peaks was observed; the number of noise peaks is the same as the harmonic reducer ratio. Therefore, the prototype system can be used to observe the extent of vibration generated when the harmonic reducer is in operation.  
Measurements were then conducted by using a rapid stable system. The radial dimension measurement results of the prototype and rapid system were similar when using the ring gauge. The difference between these results and that obtained with the optical projector is \(-0.02\) mm. Although the signal of the rapid machine had considerable noise, it could be reduced by using a Butterworth filter, and the total eccentricity error is controlled to within 0.04 mm.  
In the rapid measurement machine, due to the need for a larger sampling rate/data, the I/O unit was replaced, leading to errors in the signal noise and the sampling interval. The maximum error in tooth thickness measurement is 0.48 mm because the signal was only filtered by a simple direct filter. We will attempt to resolve this issue in our future work.  
Whether we used the prototype or rapid system, the radial measurement dimension error of the ring gauge is within \(-0.02\) mm. This means that the system is highly stable, accurate, and yields reproducible results.  
The novel rapid measurement system was compared with the CMM used in the production line. The difference in measurement between the two systems is 0.08 mm, and thus we determined that our system can be directly transferred to the production line after completion. Another good benefit is on the equipment prices: CMM is approximately USD 50,000, and the proposed system is around USD 20,000.  
In the future, to solve the problem of reducer-generated noise, the drive mechanism of the rotation platform may be changed to a direct drive motor.  
Conceptualization, Q.-C.H.; methodology, Q.-C.H.; software, Y.-L.C., X.-Q.L. and Z.-R.Y.; validation, Y.-L.C. and Z.-R.Y.; investigation, Q.-C.H.; writing--original draft preparation, Y.-L.C.; writing--review and editing, Q.-C.H.; visualization, Y.-L.C., X.-Q.L. and Z.-R.Y.; supervision, Q.-C.H.; project administration, Q.-C.H.; funding acquisition, Q.-C.H. All authors have read and agreed to the published version of the manuscript.  
This research was partially funded by the Ministry of Science and Technology, Taiwan, under grant number MOST 106-2622-E-151-017-CC3. The authors would like to thank Lineson Industrial Co., Ltd. (Taiwan) for providing the measurement specimens and for valuable advice.  
The authors declare no conflict of interest.