# A Precise Modeling Method for PMSM Servo System of Spacecraft
## 3 Nonlinear Precise Modeling
### Friction Characteristics Modeling  
Friction characteristics can be divided into static friction and dynamic friction according to their motion characteristics. In actual systems, multiple types of friction forces usually coexist, such as the Stribeck model. The actual operating state of the servo system may change dynamically in real time under different operating conditions. Considering  
Figure 7: Time-varying mesh stiffness of the gear. (**a**) Gear meshing process; (**b**) Local magnification; (**c**) Single pair gear deformation; (**d**) Stiffness superposition. the static and dynamic characteristics of friction, the LuGre friction model can comprehensively characterize the dynamic and static processes of the frictional link among various friction models and is currently one of the most widely studied and applied friction models. Compared with the Stribeck model, the LuGre model can more accurately describe the friction change under dynamic velocity. This paper characterizes the friction characteristics of spacecraft servo systems based on their application characteristics. The friction force is influenced by multiple factors and parameters, and the environment and operating conditions are complex and variable. The LuGre friction model is adopted to represent the friction characteristics of the servo system. The LuGre friction model can be expressed as  
\[\left\{ \begin{array}{l}F_{f}=\sigma_{0}z+\sigma_{1}\frac{dz}{dt}+\sigma_{2} \dot{\theta}\\ F_{zf}=\sigma_{0}z+\sigma_{1}\frac{dz}{dt}\\ F_{pf}=\sigma_{2}\dot{\theta}\\ \frac{dz}{dt}=\dot{\theta}-\frac{\left|\dot{\theta}\right|}{g(\theta)}z\\ \sigma_{0}g(\dot{\theta})=F_{c}+\left(F_{s}-F_{c}\right)\exp\left[-\left(\dot {\theta}/\dot{\theta}_{s}\right)^{2}\right]\end{array} \right. \tag{7}\]  
where \(F_{f}\) represents the LuGre friction force; \(F_{zf}\) represents the friction force associated with the mane offset; \(F_{pf}\) represents the viscous friction force related to the velocity; \(\dot{\theta}\) is the velocity of the actuator; \(\sigma_{0}\), \(\sigma_{1}\), \(\sigma_{2}\) are stiffness coefficient, damping coefficient and viscous friction coefficient, respectively; \(z\) is the average offset between the bristles; \(F_{c}\) represents Coulomb friction force; \(F_{s}\) represents maximum static friction force; \(\dot{\theta}_{s}\) is the Stribeck velocity. The model shown in Equation (7) only considers the case where the operating conditions are relatively stable and does not take into account the impact of external environmental changes. Spacecraft servo systems operate in a wide temperature range of [\(-55\)\({}^{\circ}\)C, \(180\)\({}^{\circ}\)C], while being subjected to unknown airflow disturbances such as gusts and turbulence. When spacecraft servo systems experience vibration, shock, or acceleration/deceleration, the contact between the bristles in the friction model changes, thereby affecting the magnitude of friction force. Selecting \(\mu\) as the influencing factor of the bristle offset z-related friction force \(F_{zf}\) caused by vibration and shock, where \(\mu>0\), establish its expression as  
\[F_{zf}=\mu\bigg{(}\sigma_{0}z+\sigma_{1}\frac{dz}{dt}\bigg{)} \tag{8}\]  
When the environmental temperature of the servo system changes, the lubrication state of the servo mechanism also changes with the temperature, thereby affecting the viscous friction coefficient. Therefore, introduce a temperature influence factor \(\lambda\) to represent the effect of temperature change on the viscous friction force \(F_{pf}\), where \(\lambda>0\), and establish its expression as  
\[F_{pf}=\lambda\sigma_{2}\dot{\theta} \tag{9}\]  
Combining Equations (8) and (9), the modified LuGre friction model considering environmental factors can be obtained as follows:  
\[F_{cf}=\mu\bigg{(}\sigma_{0}z+\sigma_{1}\frac{dz}{dt}\bigg{)}+\lambda\sigma_{ 2}\dot{\theta} \tag{10}\]  
Specifically, when the spacecraft servo system is in a ground test environment, the state is relatively stable, with \(\mu=\lambda=1\).