## 3 SCH Nonlinear Mathematical Model (NMM)
### Nonlinear Model Modification
#### 3.2.1. Hover Modification  
In hover condition, Configuration D model was applied in the tests. According to the previous analysis, the coupled roll pitch TPP equations (Equation (2)) were in a linear form, which can be easily modified using IDM results by  
\[ \begin{array}{l}\tau_{f}\dot{a}=-a-\tau_{f}q-A_{b}b+A_{\delta_{lon}}\delta_{ lom}+A_{\delta_{lat}}\delta_{lat}\\ \tau_{f}\dot{b}=-b-\tau_{f}p-B_{a}a+B_{\delta_{lat}}\delta_{lat}+B_{\delta_{lom} }\delta_{lon}.\end{array}  \tag{34}\]  
The flap equations in NMM were replaced entirely by Equation (34). The yaw dynamics contained a strong head-lock control law (in a 3-axis gyro component), which was linear too. It can be decided \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Hover**} & \multicolumn{3}{c}{**30 m/s Configuration D**} & \multicolumn{1}{c}{**30 m/s Configuration WD**} \\ \hline
**Parameter** & **Value** & **Parameter** & **Value** & **Parameter** & **Value** & **Parameter** & **Value** \\ \hline \(\tau_{f}\) & \(0.05560\) & \(A_{\delta_{lat}}\) & \(-0.03270\) & \(\tau_{f}\) & \(0.0510\) & \(X_{u}\) & \(-0.328\) \\ \(X_{u}\) & \(-0.06020\) & \(A_{\delta_{lon}}\) & \(-0.4568\) & \(L_{b}\) & \(331.2\) & \(L_{p}\) & \(-3.09\) \\ \(Y_{v}\) & \(-0.1420\) & \(B_{\delta_{lat}}\) & \(0.5046\) & \(B_{\delta_{lat}}\) & \(0.2680\) & \(L_{\delta_{v}}\) & \(3.53\) \\ \(Z_{w}\) & \(-1.714\) & \(B_{\delta_{lat}}\) & \(-0.05330\) & \(X_{u}\) & \(-0.297\) & & \\ \(M_{u}\) & \(448.4\) & \(Z_{\delta_{lat}}\) & \(-45.87\) & \(Y_{v}\) & \(-0.115\) & & \\ \(L_{b}\) & \(740.9\) & \(N_{pol}\) & \(75.28\) & & & & \\ \(N_{r}\) & \(-1.084\) & & & & & & \\ Cost value & & \(39.9078\) & & Cost value & \(47.7424\) & Cost value & \(58.1301\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Main parameters from linear identification model (IDM). Figure 7: Frequency domain comparisons of IDM and flight responses in hover condition. (Solid line: flight responses; Dash line: IDM results). (**a**) \(q/\delta_{lon}\); (**b**) \(p/\delta_{lat}\); (**c**) \(r/\delta_{pol}\); (**d**) \(n_{x}/\delta_{col}\). entirely by the identified gyro model [25], which contains the parameters \(N_{r}\) and \(N_{\delta_{pd}}\) in Table 4. Apart from these two directions, there are six essential derivatives to be discussed:  
1. Heave damping derivative \(Z_{w}\) and heave maneuver derivative \(Z_{col}\):  
Because, the main rotor dominated the heave direction throughout the flight envelope [19]. \(Z_{w}\) and \(Z_{col}\) can be expressed as:  
\[Z_{w}=a_{0}\cdot\frac{\rho(\Omega_{mr}R_{mr})\pi R_{mr}^{2}}{m}\cdot\frac{4s \lambda_{i}}{16\lambda_{i}+a_{0}s} \tag{35}\]  
\[Z_{\delta_{col}}=a_{0}K_{col}\cdot\frac{8}{3}\frac{\rho(\Omega_{mr}R_{mr})^{2} \pi R_{mr}^{2}}{m}\frac{s\lambda_{i}}{16\lambda_{i}+a_{0}s} \tag{36}\]  
where \(K_{col}\) is the scaling parameters to transform \(\theta_{col}(rad)\) to \(\delta_{col}([-0.5,0.5])\). In the above equations, \((\rho(\Omega_{mr}R_{mr})^{2}\pi R_{mr}^{2}/m)\) was a constant value. Assume \(\lambda_{i}\) and \(s\) were estimated correctly, then the two nonlinear parameters \(a_{0}\) and \(K_{col}\) can be updated by \(Z_{w}\) and \(Z_{\delta_{col}}\) from IDM. 2. Pitch damping derivative \(M_{a}\) and roll damping derivative \(L_{b}\):  
Similarly, the main pitch and roll moments were mainly decided by the main rotor in an approximate form:  
\[M_{a}=-(N_{b}\frac{K_{\beta}}{2}+T_{mr}H_{mr})/I_{xx} \tag{37}\]  
\[L_{b}=-(N_{b}\frac{K_{\beta}}{2}+T_{mr}H_{mr})/I_{yy} \tag{38}\]  
where \(N_{b}\) is the blade number. \((T_{mr}H_{mr})\) was a small value. Assume \((T_{mr}H_{mr})\) was estimated correctly, then the nonlinear parameters \(K_{\beta}\), \(I_{xx}\) and \(I_{yy}\) can be corrected by \(M_{a}\) and \(L_{b}\) from IDM. 3. Forward velocity derivative \(X_{u}\) and lateral velocity derivative \(Y_{v}\):  
The forward velocity derivative was influenced mainly by main rotor and airframe. \[X_{u}=X_{umr}+X_{uf} \tag{39}\]  
where \(X_{uf}\) depends on \(S_{fx}\) (the forward effective drag area of airframe).