# Modeling of Transmission Compliance and Hysteresis Considering Degradation in a Harmonic Drive
## 3 Behavioral Degradation  
In order to evaluate the behavioral degradation, a specialized harmonic drive test device was designed to observe the behavior change of the reducer. As depicted in Figure 6, it is an electromechanical system that includes a driving servo motor, a loading servo motor, two rotary encoders, two torque sensors, an HD unit, and a digital control module. The concentricity was adjusted using an Easy-Laser shaft alignment instrument. The main performance parameters of the test apparatus are displayed in Table 1. An HD (CSF-25-80) was used for the behavioral degradation test. The reduction ratio was 80:1, and it had a rated output torque of 63 Nm. To determine the torsional compliance and hysteresis behaviors of the HD experimentally, first, the input shaft of the drive was locked. Next, the load torque was controlled to vary in a linear manner, and the resulting angular displacement for the load position was  
Figure 6: View of the harmonic drive test apparatus. measured by the rotary encoder and recorded. The initial torque-displacement relationship of the reducer is shown in Figure 7. Then the reducer was operated stably for 100 h at a rated torque of 63 Nm and a rated speed of 2000 rpm, and its torque-displacement relationship was tested (Figure 7). The wear conditions of the rubbing surface of the HD unit after the test are shown in Figure 8. The wear was visible on the FS tooth surface and FS inner surface. Figure 7 shows that behavioral degradation of the reducer occurred. Therefore, in order to improve the performance of the system, it was necessary to capture the degradation behavior of the reducer. Using the analysis of the HD model in Figure 5, the transmission compliance and hysteresis in HD were mainly attributed to the flexibility of the FS cup, the FS teeth, and the WG, as well as the friction in the tooth engagement area. The degradation of the behavior of the reducer was caused by the wear of the FS tooth surface and the FS inner surface, which led to the degradation of the stiffness of the FS tooth and the friction in the tooth engagement area. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Name** & **Rating** & **Resolution** \\ \hline Driving servo motor & 6.5 Nm & - \\ Torque sensor for the input side & 20 Nm & 0.02 Nm \\ Encoder for input side & - & 0.00439\({}^{\circ}\) \\ Loading servo torque motor & 500 Nm & 0.5 Nm \\ Torque sensor for the output side & 1000 Nm & 0.1 Nm \\ Encoder for the output side & - & 0.00056\({}^{\circ}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: List of the main performance parameters of the test system. Figure 8: The condition of the rubbing surface after the test: (**a**) FS inner surface, and (**b**) FS tooth. Figure 7: The transmission compliance and hysteresis behaviors change of the HD. For the FS tooth, wear made the tooth and tooth foundation thinner, and reduced the depth of engagement and number of meshing teeth, which led to degradation of the combined stiffness of meshing tooth pairs. \[K^{\prime}{}_{T}=\left\{ \begin{array}{ll}a^{\prime}T_{FS}+b^{\prime}&T_{FS} \leq T^{\prime}{}_{T}\\ a^{\prime}T^{\prime}{}_{T}+b^{\prime}&T_{FS}>T^{\prime}{}_{T}\end{array} \right. \tag{22}\]  
At the same time, with the increase of wear, the amount of interference of the tooth surface gradually decreased, and when the wear reached a certain degree, the interference disappeared. Therefore, the no-load preload \(T_{0}\) due to inherent interference also decreased with the increase of wear until it reached 0. \[T_{0}{}^{\prime}=\left\{ \begin{array}{ll}c^{\prime}(T_{FS}-T^{\prime})&T_{FS} \leq T^{\prime}_{1}\\ 0&T_{FS}>T^{\prime}_{1}\end{array} \right. \tag{23}\]  
Equations (22) and (23) capture the degradation of this behavior in the proposed model. Figure 7 shows that the initial transmission compliance and hysteresis in HD had the symmetric property, but the reducer did not have such properties after 100 h of operation. This is most likely because the test was conducted in one direction.