# Modeling of Transmission Compliance and Hysteresis Considering Degradation in a Harmonic Drive
## 2 Mechanical Model
### Basic Model with Friction, Compliance, and Kinematic Error  
In order to capture the synthesis effects of friction, torsional compliance, and kinematic error, a model simplified by using a translational motion model analogy, was proposed by [PERSON] et al. [11] (Figure 3). In the comprehensive model, the kinematic and mechanical relationships can be expressed as follows:  
\[\theta_{WG}=-N(\theta_{FS}+\theta_{e}+\Delta\theta) \tag{6}\]  
\[T_{WG}=-\frac{1}{N}\Big{(}T_{FS}+T_{f}\Big{)} \tag{7}\]  
Figure 2: A kinematic representation of a harmonic drive showing the three ports.  
where \(\theta_{e}\) is the kinematic error, \(\Delta\theta\) is the torsional angle of the HD, and \(T_{f}\) is the friction torque. In the model, the compliance behavior is described by modeling the overall behavior of HD.  
According to the description in a manufacturer's catalog [5], the nonlinear spring can be approximated by a piecewise linear function of the torsion angle:  
\[\Delta\theta=\left\{\begin{array}{ll}\frac{T_{FS}}{K_{1}}&T_{FS}\leq T_{1} \\ \frac{T_{1}}{K_{1}}+\frac{T_{FS}-T_{1}}{K_{2}}&T_{1}\leq T_{FS}\leq T_{2}\\ \frac{T_{1}}{K_{1}}+\frac{T_{2}-T_{1}}{K_{2}}+\frac{T_{FS}-T_{2}}{K_{3}}&T_{FS} \geq T_{2}\end{array}\right. \tag{8}\]  
where \(K_{1}\), \(K_{2}\), and \(K_{3}\) are the stiffness; and \(T_{1}\) and \(T_{2}\) are the torque given by the manufacturer. Another method of estimating the nonlinear stiffness uses a cubic polynomial approximation [12]. The third-order polynomial function is given by:  
\[T_{FS}=a_{3}(\Delta\theta)^{3}+a_{1}\Delta\theta \tag{9}\]  
where \(a_{1}\) and \(a_{3}\) are constants to be determined.  
In order to estimate the friction torque, the Stribeck friction model was used to capture the nonlinear friction behavior [26]. Therefore, the friction torque \(T_{f}\) can be expressed as:  
\[T_{f}=\left[F_{\varepsilon}+(F_{\mathrm{s}}-F_{c})e^{-(\hat{\theta}_{\mathrm{ WG}}/\Omega)^{\delta}}+F_{\mathrm{v}}\hat{\theta}_{\mathrm{WG}}\right]sign \Big{(}\hat{\theta}_{\mathrm{WG}}\Big{)} \tag{10}\]  
where \(F_{\mathrm{s}}\) denotes static friction, \(F_{c}\) denotes the minimum value of the Coulomb friction, \(\Omega\) and \(F_{\mathrm{v}}\) are lubricant and load parameters, and \(\delta\) is an additional empirical parameter.  
Any deviation of the teeth from an ideal shape results in the well-known errors in gear ratio known as the kinematic error. As described in [27], the finite Fourier series can be used to describe the kinematic error:  
\[\theta_{e}=\sum_{i=1}^{n}A_{i}\sin(\omega_{i}(\theta_{FS}/2\pi)+\Phi_{i}) \tag{11}\]  
where \(A_{i}\) is the amplitude for a particular cyclical frequency, \(\omega_{i}\) is the frequency, \(\Phi_{i}\) is the phase angle for each component, an \(n\) value of 3 is adequate to describe the kinematic error profile.  
However, as shown in Figure 4, the model described in this section does not incorporate the hysteresis behavior, which results from the interaction between transmission compliance and friction. This is because the model described in this section is a mechanistic superposition model for friction, torsional compliance, and kinematic error, and does not consider the coupling effects of torsional compliance and friction. Hysteresis behavior,  
Figure 3: A schematic translation representation of the harmonic drive.  
especially in smart materials, magnetic fields, or micro-sliding friction, was dramatically observed [28,29]. A typical HD stiffness curve (Figure 4) is a hysteresis loop that is divided into upper and lower branches. In this study, a new model based on the basic operating principle of HDs is presented to capture their nonlinear torsional compliance and hysteresis behaviors.