## 1 Introduction  
In the modern industrial field, operating efficiency and precision of different types of mechanical equipment are becoming more and more important, particularly those involving transmission systems. In this regard, the cycloidal reducer is widely used in different mechanical transmission systems because of its unique benefits of high load, high efficiency, compactness, low noise, and other qualities [1]. Advanced mathematical models and computer simulation techniques are combined with mechanical analyses and tooth modifications to optimize the efficiency of cycloidal reducers. The newly designed two-stage cycloidal reducer has established an independent dynamic model, thereby verifying the dynamic balance and stability of the two-stage cycloidal reducer [2]. A tooth contact analysis model was established using Hertz contact theory, and the transmission error and reduction ratio fluctuation were analyzed, providing a certain theoretical basis for optimizing motion loss and transmission efficiency in the future [3; 4]. A meshing efficiency model of the cycloidal gear was established, and a theoretical efficiency formula was derived [5; 6; 7; 8; 9]. By increasing the short amplitude coefficient, the transmission efficiency of the reducer was improved, and the design of the cycloidal gear was optimized. A new method was proposed for the blade design of the cycloidal plate gear [10], and a new design was carried out for the structure of the cycloidal pinwheel reducer [11], reducing its size and volume and exhibiting good dynamic balance characteristics. Meanwhile, some measurement methods [12; 13; 14; 15] were proposed to meet the accuracy requirements for the cycloidal gear, which can reduce the errors in the manufacturing process and test more parameters of the cycloid reducer with high accuracy. The preceding studies provide a significant theoretical basis for optimum design and performance enhancement of cycloidal reducers. Compared with the traditional cycloidal reducer with four cycloid discs, the new cycloidal reducer with two cycloid discs is quite compact, with superior load distribution and dynamic balancing characteristics [16]. The vibration effect of the cycloid speed reducer is reduced by using flexible pads [17]. Optimization approaches aimed at minimizing size and maximizing efficiency [18] to improve the entire mechanism or upgrading the cycloidal gear through a new profile design can also improve the performance of the cycloidal reducer. The analysis of the size and efficiency of cycloidal reducers [19] contributes to investigating further research directions for cycloidal reducers. The analysis of the meshing efficiency of cycloidal gears [20], as well as the analysis of the tooth contact and load distribution, and the consideration of the effect of tooth modification can help optimize the design of the cycloidal gears. In addition, the dynamic analysis should not be neglected when investigating cycloidal reducers. The dynamic analysis of reducers can verify whether the performance of the designed new reducer fulfills the expectation. At the same time, the dynamic analysis of various transmission designs can further verify whether the designed reducer is reasonable and can help investigate the next research direction. The cycloid disc is one of the main components of the cycloidal reducer, and it is also the most critical component. Under different meshing conditions, the dynamic analysis of the stress and strain of the cycloid disk [21; 22] can determine what happens to the components in the cycloid disc during the motion process, which can help optimize the design of the cycloid disc. Additionally, a dynamic analysis with small tooth differences in the cycloidal reducer can also improve the shortcomings of the conventional design. On the other hand, measurements of the cycloid reducer can also provide useful data support for the direction of improvement.