# Modeling and Control of an Articulated Multibody Aircraft
## 4 Simulation and Results
### Linearized Aircraft Model  
Applying the Jacobian linearization approach or small disturbance theory [96], at a steady-state condition (trimmed flight), the linearized state-space model of the aircraft was obtained. Since the aim of the proposed work is to track a reference pitch angle and yaw angle, the linearized model was decoupled into longitudinal and lateral models. The same servomotor was used for abdominal pitch and yaw motion control.  
The aircraft trim data corresponding to steady cruise flight condition was estimated for an airspeed of \(V_{0}=10\) m/s and height \(H_{0}=100\) m for an undeflected abdomen. For steady cruise, the aircraft flies with constant translational and angular velocity components in the body frame, \((\dot{u},\dot{v},\dot{w},\dot{p},\dot{q},\dot{r}=0)\). Furthermore, \((v,p,r,\phi,\psi=0)\). Only the elevator \(\delta_{e}\) and thrust \(T_{n}\) were used as control inputs to trim the aircraft. MATLAB's trim algorithm that uses sequential quadratic programming was used to obtain the trim solution. The resulting trimmed steady cruise flight condition parameters are presented in Table 3. Simulation constraints on states and control inputs are given in Equation (67).  
The nonlinear model was trimmed about steady cruise flight condition \((v,p,q,r,\phi,\psi=0)\) with Matlab's trim algorithm that uses sequential quadratic programming.  
\[\begin{cases}|\alpha|\leq 20\,^{\circ},\ |\beta|\leq 30\,^{\circ},\ |p|\leq 30\,^{ \circ}/s,\ |q|\leq 30\,^{\circ}/s,\ |r|\leq 30\,^{\circ}/s\\ \delta_{e},\delta_{a},\in[-20,20]\,(^{\circ}),\ \ \ \theta_{T},\psi_{T}\in[-60,60]\,(^{ \circ})\end{cases} \tag{67}\]  
The linearization algorithm in MATLAB was also used to linearize the trimmed aircraft. The longitudinal part of the linearized model consists of the following states: \(\mathbf{x}_{long}=[u\ w\ q\ \theta]^{\prime}\), and control inputs: \(\mathbf{u}_{long}=[\delta_{e}\ \theta_{T}]^{\prime}\). The lateral-directional part consists of the following states: \(\mathbf{x}_{lat}=[v\ p\ r\ \phi\ \psi]^{\prime}\), and control inputs: \(\mathbf{u}_{long}=[\delta_{a}\ \phi_{T}]^{\prime}\). The longitudinal state and control matrices, as well as the lateral state and control matrices are given in Equations (68) and (69), respectively.  
\[A_{long}=\begin{bmatrix}-0.39&0.18&0.13&-9.81\\ -3.16&-26.62&6.82&0.26\\ -7.80&-67.54&-8.74&0.64\\ 0&1&0\end{bmatrix},\ \ \ B_{long}=\begin{bmatrix}0&0\\ -2.2&1.65\\ -6.43&-190.03\\ 0&0\end{bmatrix} \tag{68}\]  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(V_{0}\) (m/s) & \(H_{0}\) (m) & \(\theta_{T_{0}}\) (\({}^{\circ}\)) & **Model Type** & \(\theta_{0}\) (\({}^{\circ}\)) & \(\delta_{e0}\) (\({}^{\circ}\)) & \(T_{n_{0}}\) (N) & \(\tau_{asy0}\) (Nm) \\ \hline \multirow{4}{*}{10} & \multirow{4}{*}{100} & \multirow{4}{*}{100} & Single-body & \(-11.35\) & \(0.228\) & \(0.679\) & \multirow{4}{*}{-0.235} \\  & & Multibody & \(-11.35\) & \(0.228\) & \(0.679\) & \(-0.235\) \\ \cline{1-1}  & & & Single-body & \(-10.66\) & \(0.0707\) & \(0.683\) & \(-0.232\) \\ \cline{1-1}  & & & Multibody & \(-6.02\) & \(-0.96\) & \(0.708\) & \(-0.204\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Single-body vs multibody trim results at steady cruise trim condition.  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{2}{c}{**State (\(\mathbf{x_{0}}\))**} & \multicolumn{2}{c}{**Control Input (\(\mathbf{u_{0}}\))**} \\ \hline \(u_{0}=10\) (m/s) & \(p_{0}=0\) (\({}^{\circ}\)/s) & \(\delta_{e0}=0.228\) (\({}^{\circ}\)) & \(\phi_{T_{0}}=0\) (\({}^{\circ}\)) \\ \(v_{0}=0\) (m/s) & \(q_{0}=0\) (\({}^{\circ}\)/s) & \(\delta_{a0}=0\) (\({}^{\circ}\)) & \(\  
\[A_{lat}=\begin{bmatrix}-0.25&-0.32&-9.91&12.22&0\\ -14.09&-85.94&12.47&0&0\\ 0.45&-0.014&-0.49&-23.33&0\\ 0&1&-0.02&0&0\\ 0&0&1&0&0\end{bmatrix},\quad B_{lat}=\begin{bmatrix}-0.033&-7.11\\ -18.71&0\\ 0.32&-137.27\\ 0&0\\ 0&0\end{bmatrix} \tag{69}\]