# Modeling and Control of an Articulated Multibody Aircraft
## 4 Simulation and Results
### Model Validation: Single-Body vs. Multibody  
The verification/validation of the proposed multibody model presented in Sections 2.6 and 2.7 are presented in this section. The states and control inputs for the aircraft considered are:  
\[\mathbf{x}=(u,v,w,p,q,r,\phi,\theta,\phi,X,Y,Z)^{T}, \tag{63}\]  
\[\mathbf{u}=(\delta_{e},\delta_{a},T_{n},\phi_{T},\theta_{T},\psi_{T})^{T}, \tag{64}\]  
The single-body and multibody model of the same aircraft should present the same longitudinal trim results. The translational and rotational flight dynamics equations of motion for a single rigid body not referenced to the center of mass have been derived by [PERSON] and [PERSON] in [134] as  
\[m\left([\dot{V}_{b}^{I}]^{B}+[\Omega^{BI}]^{B}[V_{b}^{I}]^{B} \right)+\left([\Omega^{BI}]^{B}m[s_{cb}]^{B}\right) \tag{65}\] \[+\left([\Omega^{BI}]^{B}\left([\Omega^{BI}]^{B}m[s_{cb}]^{B} \right)\right)=[F]^{B}\] \[\left(m[S_{cb}]^{B}[\dot{V}_{b}^{I}]^{B}\right)+\left(m[\Omega^{ BI}]^{B}([S_{cb}]^{B}[V_{b}^{I}]^{B})\right)\] \[\left(m[V_{b}^{I}]^{B}([\Omega^{BI}]^{B}[S_{cb}]^{B})\right)+[I_{ b}^{C}]^{B}[\dot{\omega}^{BI}]^{B}\] (66) \[+\left([\Omega^{BI}]^{B}[I_{b}^{C}]^{B}[\omega^{BI}]^{B}\right)= [M]^{B}+\left[[S_{cb}]^{B}[F]^{B}\right]\]  
The longitudinal trim results of the single-body model in [134] and the multibody model presented in this paper are calculated and compared. For the single-body model, the aircraft rotation of the abdomen was not considered during the estimation for longitudinal trim. The longitudinal trim using the single-body model can be described as follows: for a given airspeed \(V\) and a fixed abdominal pitch angle \(\theta_{T}\), the objective is to estimate the appropriate longitudinal pitch angle \(\theta\) and controls elevator \(\delta_{e}\) and thrust \(T_{n}\), to make the longitudinal rates, \(\dot{u},\dot{w},\dot{q}\) and \(\dot{h}\) approach zero.  
In the case of the multibody model, the longitudinal trim can be described as follows: for a given airspeed \(V\) and a fixed abdominal pitch angle \(\theta_{T}\), the objective is to estimate the appropriate longitudinal pitch angle \(\theta\) and controls, elevator \(\delta_{e}\) and thrust \(T_{n}\), required to keep the abdomen pitch angle \(\theta_{T}\) at a given value to make the longitudinal rates \(\dot{u},\dot{w},\dot{q},\dot{h}\) and \([\dot{\omega}^{TB}]^{T}\) approach zero. Because the abdominal motion dynamics are included in the multibody model, the torque \(\tau_{ay}\) required to keep the abdominal pitch inclination angle at the specified value can also be estimated.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline \(m^{B}\) (kg) & 0.325 & \(m^{T}\) (kg) & 0.06 \\ Body length, \(l_{B}\) (m) & 0.3 & Tail length, \(l_{T_{max}}\) (m) & 0.4 \\ Max. body diameter, \(d_{B_{max}}\) (m) & 0.14 & Tail diameter, \(d_{T}\) (m) & 0.05 \\ \(l_{xx}^{B}\) (kg\(\cdot\)m\({}^{2}\)) & 0.00187 & \(b_{ref}\) (m) & 1.4 \\ \(l_{yy}^{B}\) (kg\(\cdot\)m\({}^{2}\)) & 0.01117 & \(c_{ref}\) (m) & 0.19434 \\ \(l_{zz}^{B}\) (kg\(\cdot\)m\({}^{2}\)) & 0.00934 & \(S_{ref}\) (m\({}^{2}\)) & 0.26865 \\ \(c_{S}^{B}\) (m) & \([-0.064;\) 0; 0.003] & \(ARP\) (m) & [0.025; 0; 0] \\ \hline \hline \end{tabular}
\end{table}
Table 1: Aircraft model physical properties [131].  
The longitudinal trim problem is solved in MATLAB for an airspeed \(V_{0}=10\) m/s, a height \(H_{0}=100\) m and abdominal pitch inclination angles \(\theta_{T_{0}}=0,-10\) and \(-30^{\circ}\). The results are presented in Table 2.