# A New Type Bionic Foldable Wing Design for High Maneuverable Unmanned Aerial Vehicle
## 2 Preliminary Design of the Folding Wing Structure
### Dynamics Model of Fold-Wing Aircraft
#### 2.1.3 Rotational Dynamics  
For the i-th wing, Equation (6) can be obtained according to the definition of moment of momentum:  
\[\mathbf{H_{i}}=\mathbf{\rho_{i}}\times\sum\delta\mathrm{m_{i}}\bigg{(}\mathbf{ v_{0}}+\frac{\mathrm{d}\mathbf{\rho_{i}}}{\mathrm{d}\mathbf{t}}\bigg{)} \tag{6}\]  
Take the derivative of both sides and obtain Equation (7):  
\[\frac{\mathrm{d}\mathbf{H_{i}}}{\mathrm{d}\mathbf{t}}=\frac{\mathrm{d}\mathbf{ \rho_{i}}}{\mathrm{d}\mathbf{t}}\times\sum\delta\mathrm{m_{i}}\mathbf{v_{0}}+ \mathbf{\rho_{i}}\times\sum\delta\mathrm{m_{i}}\frac{\mathrm{d}\mathbf{\rho_{i }}}{\mathrm{d}\mathbf{t}}+\rho_{i}\times\sum\delta\mathrm{m_{i}}\frac{\mathrm{d }^{2}\mathbf{\rho_{i}}}{\mathrm{d}\mathbf{t}^{2}} \tag{7}\]  
\[\frac{\mathrm{d}\mathbf{H_{i}}}{\mathrm{d}\mathbf{t}}=\sum\delta\mathrm{m_{i}} \mathbf{\rho_{i}}\times\frac{\mathrm{d}\mathbf{v_{i}}}{\mathrm{d}\mathbf{t}}+ \sum\delta\mathrm{m_{i}}\frac{\mathrm{d}\mathbf{\rho_{i}}}{\mathrm{d}\mathbf{t}} \times\mathbf{v_{i}}=\mathbf{M_{oi}}+\sum\delta\mathrm{m_{i}}\frac{\mathrm{d }\mathbf{\rho_{i}}}{\mathrm{d}\mathbf{t}}\times\mathbf{v_{i}}=\mathbf{M_{oi}}+ \mathrm{m_{i}}\frac{\mathrm{d}\mathbf{\rho_{i}}}{\mathrm{d}\mathbf{t}}\times \mathbf{v_{o}} \tag{8}\]  
**M\({}_{\mathbf{oi}}\)**--the external torque generated by the external force acting on the i-th folding wing on the origin of the coordinate system.  
\[\mathbf{M_{oi}}=\mathrm{m_{i}}\mathbf{\rho_{i}}\times\frac{\mathrm{d}\mathbf{v_ {o}}}{\mathrm{d}\mathbf{t}}+\mathrm{m_{i}}\mathbf{\rho_{i}}\times\frac{ \mathrm{d}^{2}\mathbf{\rho_{i}}}{\mathrm{d}\mathbf{t}^{2}} \tag{9}\]According to the above equation, write Equation (10) for each rigid body:  
\[\left\{\begin{array}{c}\mathbf{M_{af}}+\mathbf{M_{1f}}+\mathbf{M_{2f}}=\frac{ \mathbf{d}\mathbf{H_{f}}}{\mathrm{d}\mathbf{f}}\\ \mathbf{M_{a1}}+\mathbf{M_{G_{i}}}+\mathbf{M_{f1}}=\mathrm{m_{i}}\mathbf{\rho_ {i}}\times\frac{\mathrm{d}\mathbf{v_{a}}}{\mathrm{d}\mathbf{t}}+\mathrm{m_{i}} \mathbf{\rho_{i}}\times\frac{\mathrm{d}^{2}\mathbf{\rho_{i}}}{\mathrm{d}\mathbf{ t}^{2}}\\ \mathbf{M_{a2}}+\mathbf{M_{G_{2}}}+\mathbf{M_{Z}}=\mathrm{m_{2}}\mathbf{\rho_ {2}}\times\frac{\mathrm{d}\mathbf{v_{a}}}{\mathrm{d}\mathbf{t}}+\mathrm{m_{2} }\mathbf{\rho_{2}}\times\frac{\mathrm{d}^{2}\mathbf{\rho_{2}}}{\mathrm{d}\mathbf{ t}^{2}}\end{array}\right. \tag{10}\]  
The above equations are collated to obtain Equation (11):  
\[\mathbf{M_{a}}+\sum_{i=1}^{2}\mathbf{\rho_{i}}\times\mathrm{m_{i}}\mathbf{g}- \sum_{i=1}^{2}\left(\mathrm{m_{i}}\mathbf{\rho_{i}}\times\frac{\mathrm{d} \mathbf{v_{a}}}{\mathrm{d}\mathbf{t}}+\mathrm{m_{i}}\mathbf{\rho_{i}}\times \frac{\mathrm{d}^{2}\mathbf{\rho_{i}}}{\mathrm{d}\mathbf{t}^{2}}\right)=\frac{ \mathrm{d}\mathbf{H_{f}}}{\mathrm{d}\mathbf{t}} \tag{11}\]  
The aerodynamic torque applied to the aircraft is the left part of Equation (11) minus \(\mathbf{M_{a}},\mathbf{M_{a}}=\left[\mathbf{M_{x}}\ \ \mathbf{M_{y}}\ \ \mathbf{M_{z}}\right]^{\mathrm{T}}\).What is left is the additional moment \(\Delta\mathbf{M}\). In the formula, \(\mathbf{M_{x}}\)\(\mathbf{M_{y}}\) and \(\mathbf{M_{z}}\) are rolling moments, pitching moments, and yawing moments, respectively.  
Equations (4) and (11) are the dynamics models of foldable wing aircraft. The aircraft is the same as the aircraft with a conventional layout in the case of a symmetrical wing, and the added torque is zero.  
When the morphing wing moves asymmetrically, wing folding will increase the additional force \(\mathbf{\Delta F}\) and torque \(\Delta\mathbf{M}\). The generation of force \(\mathbf{\Delta F}\) and moment \(\Delta\mathbf{M}\) is related to changes in the center of mass and moment of inertia.