## 1 Introduction
The cycloid reducer's tooth profile error and correction are critical parameters that affect the cycloidal reducer, and the measurement of the whole tooth profile error through radial tracking mode in polar coordinates can reduce the influence of other errors on the correction of cycloidal gear. The centering angle error of the gear shaft plays a significant role in the measuring results of flank profile and tooth rail deviation in the precision measurement of the cycloidal gear tooth surface, and it can help increase the measuring precision of cycloidal gears by compensating the centering angle inaccuracy of the gear shaft contained in the measurement results. For manufacturing errors, digital measurement of cycloidal gear errors in production based on the gear measuring center can offer the foundation for precise assessment and efficient management of cycloidal gear-producing accuracy. In the current situation where there is no design standard or test platform verification method for cycloid reducers, the cycloid reducer integrated and separated test platform can test more performance characteristics with high precision and exhibit integrated test performance, which aids in defining the design standard for the cycloid reducer. Quasi-static analysis and tooth profile modification optimization, which are crucial components in the design of cycloidal reducers, seek to improve transmission accuracy, decrease noise, and increase operational life. The quasi-static force of the cycloid reducer is thoroughly examined in this paper. Quasi-static force is defined as the magnitude and change rule of the load applied to each component during cycloid reducer movement. It is possible to gain a deeper understanding of the cycloid reducer's operating state under various working situations, to precisely estimate its performance and lifespan, and to provide the theoretical framework for the correction of tooth profile by analyzing the quasi-static force. In the quasi-static analysis, advanced mechanical models and calculation methods were utilized to conduct in-depth research on the load distribution and variation patterns of the cycloidal reducer under various operating conditions. This includes but is not limited to different types of loads, such as static load, dynamic load, impact load, and so on. At the same time, the influence of material properties, manufacturing errors, assembly errors, and other factors were also taken into account to realize a comprehensive and accurate analysis of quasi-static force. Moreover, the correction method of the tooth profile was also investigated. The tooth profile has a significant impact on the transmission efficiency, noise, vibration, and life of cycloidal reducers. Therefore, reasonable correction of tooth profile can not only reduce contact stress between gears and avoid tooth wear and fracture, but it can also improve gear meshing qualities, reduce vibration and noise, and increase transmission efficiency. In terms of tooth profile correction, modern manufacturing technology and data analysis were combined to study how to accurately control the generation and correction of tooth profiles. In addition, advanced calculation methods were used to assess the performance of the modified flank profile and optimize its design. In conclusion, quasi-static analysis and tooth profile correction of cycloid reducers is a systematic and complex scientific problem that involves the knowledge of several disciplines, such as mechanics, manufacturing, and material science. Furthermore, this issue is motivated by demands from real-world application scenarios in addition to theoretical difficulties. The cycloid reducer is an important component in both scientific research and industrial production. As a result, the optimization and improvement regarding its performance directly affect the quality and efficiency of many industrial products.