# A Precise Modeling Method for PMSM Servo System of Spacecraft
## 4 Experimental Testing and Validation
### Nonlinear Identification
#### 4.1.3 LuGre Friction Parameter Identification  
Friction parameter identification is performed based on the LuGre friction model formulas (7) and (10). Since the state is relatively stable in a ground test environment, \(\mu=\lambda=1\) is taken. The static parameters \(F_{c}\), \(F_{s}\), \(\dot{\theta}_{s}\), and \(\sigma_{2}\) of the LuGre model can be  
Figure 10: Backlash of the servo system.  
Figure 9: Dynamic stiffness of the servo system.  
identified through offline experiments. When the servo system is operating in a steady-state, we have \(\dot{z}=0\), and according to Formula (7), the friction force under steady-state can be obtained as follows:  
\[F_{sf}=\bigg{\{}F_{c}+(F_{s}-F_{c})\exp\left[-(\dot{\theta}/\dot{\theta}_{s})^{2 }\right]\bigg{\}}\text{sgn}(\dot{\theta})+\sigma_{2}\dot{\theta} \tag{13}\]  
By measuring the \(q\)-axis current signal of the servo system at steady-state operating speeds multiple times, multiplying the current by the torque coefficient yields the corresponding torque. The velocity-torque curve is fitted using the least squares method, and finally, the steady-state friction parameters are obtained according to Equation (13). The experiment measured the motor torque changes at different speeds from \(1^{\circ}\text{-s}^{-1}\) to \(10^{\circ}\text{-s}^{-1}\) in the forward and reverse directions, as shown in Figure 11. The friction model curve is fitted according to the measured points, as shown by the red line in the figure. The RMSE (Root Mean Squared Error) of the data points and fitted values is calculated to be \(1.415\times 10^{-3}\). The static parameters calculated by Formula (13) are presented in Table 1.  
When the system is in critical or low-speed motion, it is mainly affected by Coulomb friction. At this time, there is an approximate relationship where \(z\approx\theta\) and \(dz/dt\approx\dot{\theta}\). Substituting these values into Formula (7) yields:  
\[F_{df}=\sigma_{0}\theta+(\sigma_{1}+\sigma_{2})\dot{\theta} \tag{14}\]  
Based on the specific characteristics of the servo system, and the dynamic equations involving electromagnetic torque, load torque, and moment of inertia, the following Laplace transform can be obtained:  
\[\frac{\theta_{m}(s)}{i(s)}=\frac{K_{t}}{J_{m}s^{2}+(\sigma_{1}+\sigma_{2})s+ \sigma_{0}} \tag{15}\]  
where \(K_{t}\) is the torque coefficient. By setting \(\omega_{n}=\sqrt{\sigma_{0}/J_{m}}\) and \(\bar{\zeta}=\frac{\sigma_{1}+\sigma_{2}}{2\sqrt{\sigma_{0}}/\mu}\), we can obtain the transfer function of the second-order system:  
\[\frac{\theta_{m}(s)}{i(s)}=\frac{\omega_{n}^{2}}{s^{2}+2\xi\omega_{n}s+\omega _{n}^{2}}\frac{K_{t}}{\sigma_{0}} \tag{16}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{Static parameters} & \(F_{s}\)/N\(\cdot\)m & \(F_{c}\)/N\(\cdot\)m & \(\dot{\theta}_{s}\)/rad\(\cdot\)s\({}^{-1}\) & \(\sigma_{2}\)/N\(\cdot\)m\(\cdot\)rad\({}^{-1}\) \\ \cline{2-5}  & 0.024 & 0.011 & 3.73 & \(7.53\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Static parameters of LuGre friction model.  
Figure 11: Fitting curve of [PERSON] friction model.  
According to the unit step response characteristics of the second-order system  
\[\left\{\begin{array}{l}x_{0}(t)=1-\frac{\exp(-\xi\omega_{0}t)}{\sqrt{1-\xi^{2}}} \sin\left(\omega_{d}t+\arctan\frac{\sqrt{1-\xi^{2}}}{\xi}\right)\\ \omega_{d}=\omega_{n}\sqrt{1-\xi^{2}}\end{array}\right. \tag{17}\]  
where \(x_{0}(t)\) represents the output of the unit step response at time \(t\).  
By inputting a low-amplitude current step signal and collecting the response curve, \(\omega_{n}\) and \(\xi\) can be obtained, and then the values of \(\sigma_{0}\) and \(\sigma_{1}\) can be calculated. The dynamic parameters obtained from the experiment are shown in Table 2.