# Robot Joint Vibration Suppression Method Based on Improved ADRC
## 2 Mathematical Model of Robot Joint Module  
### Structure of Harmonic Reducer  
As shown in Figure 2, the harmonic reducer is mainly composed of three components: wave generator, flexible gear, and rigid gear [13]. According to the working principle of harmonic reducers, the meshing effect between the flexible gear and the wave generator can be represented by Fourier series as follows [14; 15]:  
\[k_{fc}(t)=k_{fc}+\sum\limits_{i=1}^{\infty}k_{i}\cos(2\pi if_{c}t+\psi_{i}) \tag{2}\]  
where \(f_{c}\) is the characteristic frequency of the meshing stiffness of the harmonic reducer, \(i\) is the order of the Fourier series, \(k_{i}\) is the amplitude corresponding to each Fourier series, \(\psi_{i}\) is the phase corresponding to each Fourier series, and \(k_{fc}\) is the meshing stiffness of the harmonic reducer under constant load. In the robot joint module, the wave generator is the input part of the harmonic reducer, driven by a PMSM, with an elliptical shape and a center offset. Therefore, the wave generator generates a periodic fluctuation during rotation. In the OXY coordinate system with the harmonic reducer as the origin, the meshing function between the flexible gear and the wave generator can be expressed as follows:  
\[\left\{ \begin{array}{l}k_{wfx}(t)=k_{x0}+\sum\limits_{i=2,4,6...}^{\infty}k_ {xi}\cos(2\pi if_{n}+\psi_{xi})\\ k_{wfg}(t)=k_{y0}+\sum\limits_{i=2,4,6...}^{\infty}k_{yi}\cos(2\pi if_{n}+\psi _{yi})\\ k_{wft}(t)=k_{t0}+\sum\limits_{i=2,4,6...}^{\infty}k_{ti}\cos(2\pi if_{n}+\psi _{ti})\end{array} \right. \tag{3}\]  
where \(k_{wfx}(t)\), \(k_{wfg}(t)\), \(k_{wft}(t)\) are the coupling effect of the x direction, y direction, and rotation direction, \(f_{n}\) is the rotational frequency of the PMSM, \(\psi_{xi}\), \(\psi_{yi}\), \(\psi_{ti}\) are the phases corresponding to Fourier series, \(k_{x0}\), \(k_{y0}\), \(k_{10}\) are the direct components, and \(k_{xi}\), \(k_{yi}\), \(k_{ti}\) are the amplitudes, respectively. The equation is a periodic function with the rotational frequency of the double-wave generator as the fundamental frequency. The dynamic equation of the wave generator can be described as  
\[\left\{ \begin{array}{l}m_{w}\ddot{x}_{w}+c_{wx}\dot{x}_{w}+k_{wx}x_{w}=-k_{ wfx}(t)(x_{w}-x_{f})-c_{wx}(\dot{x}_{w}-\dot{x}_{f})\\ m_{w}\ddot{y}_{w}+c_{wy}\dot{y}_{w}+k_{wy}\dot{y}_{w}=-k_{wfg}(t)(y_{w}-y_{f})- c_{wy}(y_{w}-\dot{y}_{f})\\ m_{w}\ddot{\theta}_{w}+c_{wft}\dot{\theta}_{w}+k_{wft}\theta_{w}=T_{motor}(t)-k_{wft}(t)(\theta_{w}-\theta_{f})-\\ c_{wft}(\dot{\theta}_{w}-\dot{\theta}_{f})\end{array} \right. \tag{4}\]  
where \(m_{w}\) is the quality of the wave generator, \(c_{wx}\), \(c_{wy}\), \(c_{wft}\) are the damping of the wave generator in the x direction, y direction, and rotation direction, respectively, \(k_{wx}\), \(k_{wy}\), \(k_{wt}\) are the stiffness of the wave generator in the x direction, y direction, and rotation direction,  
Figure 2: Harmonic reducer structure. respectively, \(x_{w}\), \(x_{f}\) are the vibration displacement of the wave generator in the x and y directions, \(y_{w}\) and \(y_{f}\) are the vibration displacements of the flexible gear in the x and y directions, \(\theta_{w}\) and \(\theta_{f}\) are the vibration displacements of the rigid gear in the x and y directions, and \(T_{motor}(t)\) is the output torque of the PMSM of the joint module. The dynamic equation of the flexible gear can be described as  
\[\left\{