# A Precise Modeling Method for PMSM Servo System of Spacecraft
## 3 Nonlinear Precise Modeling
### Backlash Characteristic Modeling  
The commonly used backlash models in the control field include the oscillatory model, hysteresis model, and dead-zone model. In electromechanical servo systems, the dead-zone model is widely used to describe the effect of backlash characteristics on the transmission torque. The dead-zone characteristics of the backlash are shown in Figure 5. When the  
Figure 4: Block diagram of two-mass elastic system.  
Figure 3: Two-mass elastic model.  
commutation angle is less than \(b\), the presence of the backlash will cause the transmission torque to be 0, and the original torque will be maintained at other times.  
The expression of the dead-zone model is  
\[T_{s}=\begin{cases}K_{s}(\Delta\theta-b)+B_{s}\Delta\dot{\theta},&\Delta\theta>b \\ 0,&|\Delta\theta|\leq b\\ K_{s}(\Delta\theta+b)+B_{s}\Delta\dot{\theta},&\Delta\theta<-b\end{cases} \tag{3}\]  
where \(T_{s}\) is the transmission torque; \(K_{s}\) is the equivalent stiffness of the transmission link; \(B_{s}\) is the damping coefficient of the transmission link; \(\Delta\theta=\theta_{m}-\theta_{L}\), \(b\) is the half-width of the backlash.  
Due to the small damping in the system, its influence can be neglected. Combining Equations (2) and (3), the Laplace transform of the elastic-dead-zone model can be obtained as  
\[\left\{\begin{array}{l}J_{m}\theta_{m}s^{2}=T_{e}-T_{s}\\ J_{L}\theta_{L}s^{2}=T_{s}-T_{L}\\ T_{s}=\left\{\begin{array}{l}K_{s}(\theta_{m}-\theta_{L}-b),\theta_{m}- \theta_{L}>b\\ 0,&-b<\theta_{m}-\theta_{L}<b\\ K_{s}(\theta_{m}-\theta_{L}+b),\theta_{m}-\theta_{L}<-b\\ \omega_{m}=\theta_{m}s\\ \omega_{L}=\theta_{L}s\end{array}\right.\end{array}\right. \tag{4}\]  
From this, the block diagram of the two-mass elastic system with backlash can be obtained as shown in Figure 6.  
Figure 5: Dead-zone characteristics of the backlash.  
Figure 6: Block diagram of the two-mass elastic system with backlash.  
Since the equivalent stiffness \(K_{\text{s}}\) directly affects the resonance frequency of the system, the calculation of the equivalent stiffness \(K_{\text{s}}\) is also crucial in modeling. The meshing process of the transmission system is shown in Figure 7a, and its local magnification is shown in Figure 7b. Within the entire meshing region, there will be one single-tooth meshing area and two double-tooth meshing areas. At any given moment during the meshing process, as shown in Figure 7c, the total deformation of a pair of meshed gears is the sum of the elastic deformations of the driving and driven gears, given by  
\[\delta_{\text{s}}=\delta_{p}+\delta_{g} \tag{5}\]  
Based on the theoretical relationship between stiffness and elastic deformation, and Equation (5), the expression for the stiffness of single-tooth meshing is obtained as  
\[K_{\text{s}}=\frac{1}{\delta_{s}}=\frac{1}{\delta_{p}+\delta_{g}}=\frac{k_{p}k _{g}}{k_{p}+k_{g}} \tag{6}\]  
The symbols \(\delta_{p}\) and \(\delta_{g}\) represent the deformations of the driving and driven gears, while \(k_{p}\) and \(k_{g}\) represent the stiffness of the driving and driven gears, respectively.  
According to the principle of stiffness superposition, the stiffness \(K_{ms}\) of double-tooth meshing can be obtained by superimposing the stiffness of single-tooth meshing, as shown in Figure 7d. When transitioning from single-tooth meshing to double-tooth meshing, the stiffness undergoes a step change. The meshed gears are in a state of separation and contact repeatedly during the transmission process, making the deformation of the gears exhibit strong nonlinear characteristics.