# Modeling of Transmission Compliance and Hysteresis Considering Degradation in a Harmonic Drive
## 4 Parameter Identification  
The steady-state hysteresis loop can be defined by three curves [8]: the stiffness curve, an ascending curve denoted for increasing load (d\(T_{FS}\)/d\(t\)\(>\)0), and a descending curve denoted for decreasing load (d\(T_{FS}\)/d\(t\)\(<\)0). According to the model proposed in this paper, the stiffness curve can be obtained by using the following expression:  
\[\Delta\theta=\left(\frac{1}{K_{F}}+\frac{\tan^{2}(\alpha)}{K_{W}}\right)T_{FS} +\Delta\theta_{T}(T_{FS}) \tag{24}\]  
From the analysis of the HD model in Figure 5, the torque-displacement relationship of the HD gear can be represented by two functions. For an increasing load (d\(T_{FS}\)/d\(t\)\(>\)0), the function will take the form:  
\[\Delta\theta_{u}=\frac{T_{FS}}{K_{F}}+\Delta\theta_{T}(T_{FS})+\tan\alpha \cdot\Delta\theta_{Wu} \tag{25}\]  
\[\Delta\theta_{Wu}=\left\{ \begin{array}{ll}-\frac{\mu(T_{0}-T_{FS}\cos\alpha )-T_{FS}\sin\alpha}{K_{W}(\cos\alpha-\mu\sin\alpha)}&T_{FS}\leq\frac{T_{0}\mu} {\sin\alpha+\mu\cos\alpha}\\ 0&\frac{T_{0}\mu}{\sin\alpha+\mu\cos\alpha}<T_{FS}\leq\frac{T_{0}\mu}{\sin \alpha-\mu\cos\alpha}\\ \frac{T_{FS}(\sin\alpha+\mu\cos\alpha)-T_{0}\mu}{K_{W}(\cos\alpha+\mu\sin\alpha )}&T_{FS}>\frac{T_{0}\mu}{\sin\alpha-\mu\cos\alpha}\end{array} \right. \tag{26}\]  
For a decreasing load (d\(T_{FS}\)/d\(t\)\(<\)0), the function will take the form:  
\[\Delta\theta_{d}=\frac{T_{FS}}{K_{F}}+\Delta\theta_{T}(T_{FS})+\tan\alpha \cdot\Delta\theta_{Wd} \tag{27}\]  
\[\Delta\theta_{Wd}=\left\{ \begin{array}{ll}\frac{T_{max}(\sin\alpha-\mu\cos \alpha)-T_{0}(T_{max})\mu}{K_{W}(\cos\alpha+\mu\sin\alpha)}&T_{FS}\geq\frac{K_{ W}\Delta\theta_{Wmax}(\cos\alpha-\mu\sin\alpha)-T_{0}\mu}{\sin\alpha+\mu\cos\alpha}\\ \frac{T_{FS}(\sin\alpha+\mu\cos\alpha)+T_{0}\mu}{K_{W}(\cos\alpha-\mu\sin\alpha )}&T_{FS}<\frac{K_{W}\Delta\theta_{Wmax}(\cos\alpha-\mu\sin\alpha)-T_{0}\mu} {\sin\alpha+\mu\cos\alpha}\end{array} \right. \tag{28}\]  
According to the model proposed in this paper, the nonlinearity of transmission compliance is caused by the FS teeth. When \(T_{FS}\)\(>\)\(T_{T}\), the rigidity of the FS tooth is constant, the transmission flexibility of the reducer is linear. Therefore, the threshold value \(T_{T}\) can be obtained by the method shown in Figure 9. A trial and error method can be used to estimate the parameters of the proposed HD model. The \(\mu\) and \(cT_{i}\) take initial values as \(\mu_{0}\) and \((cT_{i})_{0}\). As shown in Figure 9, five points (\(A_{i}\) (\(A_{i1}\), \(A_{i2}\)) \(i\) = 1,..., 5) are selected for parameter evaluation. Considering the test error, the average value can be obtained through several tests. Model parameters can be estimated as follows:  
\[\hat{K}_{W}=\frac{\mu_{0}(cT_{i})_{0}\tan\alpha}{A_{12}(\cos\alpha-\mu_{0}\sin \alpha)} \tag{29}\]  
\[\hat{c}=\frac{A_{32}\hat{K}_{W}(\cos\alpha-\mu_{0}\sin\alpha)}{2\mu_{0}A_{31} \tan\alpha}+\frac{(cT_{i})_{0}}{A_{31}} \tag{30}\]  
\[\hat{T}_{i}=\frac{(cT_{i})_{0}}{\hat{c}} \tag{31}\]  
\[\frac{A_{41}}{\hat{K}_{F}}+\Delta\hat{\theta}_{T}(A_{41})+\tan\alpha\Delta \hat{\theta}_{Wu}(A_{41})=A_{42} \tag{32}\]  
\[\frac{A_{51}}{\hat{K}_{F}}+\Delta\hat{\theta}_{T}(A_{51})+\tan\alpha\Delta \hat{\theta}_{Wu}(A_{51})=A_{52} \tag{33}\]  
\[\frac{A_{21}}{\hat{K}_{F}}+\Delta\hat{\theta}_{T}(A_{21})+\tan\alpha\Delta \hat{\theta}_{Wu}(A_{21})=0 \tag{34}\]  
Based on estimated stiffness parameters (\(K_{F}\), \(K_{W}\), \(a\), \(b\), \(T_{T}\)) and hysteresis parameters (\(c\), \(T_{i}\), \(\mu\)), the evaluation errors \(\varepsilon_{uj}\)\(\varepsilon_{uj}\) can be expressed as follows:  
\[\Delta\theta_{uj}=\Delta\hat{\theta}_{uj}+\varepsilon_{uj} \tag{35}\]  
\[\Delta\theta_{dj}=\Delta\hat{\theta}_{dj}+\varepsilon_{dj} \tag{36}\]  
where \(\Delta\theta_{uj}\) and \(\Delta\theta_{uj}\) are the measured hysteresis torsional angles.