# Backstepping Adaptive Neural Network Control for Electric Braking Systems of Aircrafts
## 2 Mathematical Model of the EMA
### Deformation of the Reduction Gear  
As seen from Equation (10), the relationship between the braking pressure \(p\) and the rotation angle of motor \(\theta_{m}\) is established by a linear function. However, the ideal EMA mathematical model ignored many nonlinear factors, such as the deformation of the gear and gear clearance. As described in the literature [25; 26], the reduction gear will exhibit deformation under a heavy load which can eventually affect the relationship between the braking pressure and the motor position, rendering it different from the ideal model. After many experiments, the actual curves of the braking pressure and the motor position are drawn and the comparison of the ideal and actual curves is shown in Figure 2.  
It was assumed that the deformation of the reduction gear under the braking pressure \(p\) is \(\delta_{g}\), and \(\delta_{g}\) is divided into two parts: the contact deformation \(\delta_{g1}\) and the bending deformation \(\delta_{g2}\). When analyzing the contact deformation of the reduction gear \(\delta_{g1}\), it can be simplified as a contact force model of two cylinders with parallel axes. If the contact force is small, then the two cylinders can be thought of as touching along a line parallel to their axes. As the contact force increases, the contact line deforms into a contact surface along with the elastic deformation of the material. The interface width \(w_{g}\) can be determined by the gear parameters, and the interface length \(l_{g}\) can be determined from the [PERSON] theory [27]:  
\[l_{g}=1.596\sqrt{f_{g}\cos\alpha_{g}\rho_{g}(1-\mu_{g}^{2})/E_{g}}, \tag{12}\]  
where \(\alpha_{g}\) is the angle between the acting force \(f_{g}\) and the \(y\) axis,\(\rho_{g}\) is the radius of curvature of the contact surface,\(\mu_{g}\) and \(E_{g}\) are the elastic constants of the reduction gear. Then, the stress in the reduction gear can be expressed as:  
\[\sigma_{g}=\frac{f_{g}}{l_{g}w_{g}}. \tag{13}\]  
The contact deformation of the reduction gear can be calculated using [PERSON]'s law:  
\[\delta_{g1}=\frac{f_{g}p_{g}}{l_{g}w_{g}E_{g}} \tag{14}\]  
For a fixed reduction gear, all of the parameters are constants except for the acting force \(f_{g}\). Then, Equation (14) can be simplified as:  
\[\delta_{g1}=c_{g1}\sqrt{f_{g}}, \tag{15}\]  
where \(c_{g1}\) is a constant, which is determined by the reduction gear parameters. When analyzing the bending deformation of the reduction gear \(\delta_{g2}\), the research should focus on the deformation of the meshing point so the distributed force along the tooth surface is simplified into  
Figure 2: The ideal and actual curves between the braking pressure and the motor position. a concentrated force, and the reduction gear is regarded as a cantilever of the elastic material. The force component in the \(y\) axis of the acting force \(f_{g}\) is represented as \(f_{gy}\):  
\[f_{gy}=f_{g}\cos\alpha_{g}. \tag{16}\]  
The bending deformation of the reduction gear under \(f_{gy}\) in the \(y\) axis is expressed as:  
\[\delta_{gy}=\frac{r_{g}f_{g}\cos^{2}\alpha_{g}}{E_{g}I_{g}}\int\limits_{0}^{t} \Delta x_{g}^{2}t\cos tdt, \tag{17}\]  
where \(r_{g}\) is the radius of the gear base,\(\alpha_{g}\) is the angle between \(f_{g}\) and the \(y\) axis, \(E_{g}\) is the elasticity modulus of the gear, \(I_{g}\) is the section modulus of the gear and \(\Delta x_{g}\) is the arm length of the equivalent bending moment. Then, the bending deformation of the gear is generated as:  
\[\delta_{g2}=\delta_{gy}\cos\alpha_{g}. \tag{18}\]  
Similarly, all of the parameters are constants except for the acting force \(f_{g}\). Equations (17) and (18) can be simplified as:  
\[\delta_{g2}=c_{g2}f_{g}, \tag{19}\]  
where \(c_{g2}\) is a constant, which is determined by the reduction gear parameters. After the deformation analysis, the relationship between the braking pressure \(p\) and the acting force \(f_{g}\) is needed.