# Mini Glider Design and Implementation with Wing-Folding Mechanism
## 2 Mathematical Model
### The Dynamic Model of the Glider  
The primary goal of previous calculations is to determine dynamic behaviour of the glider. To obtain the simulation of the system, a mathematical model has to be derived by Newton motion laws with six degreed of freedom (DOF) which is illustrated in Figure 3[27]. Nonlinear equations could yield  
\[X-mg\sin(\theta) = m(\dot{U}+QW-RV), \tag{6}\] \[Y+mg\cos(\theta)\sin(\phi) = m(\dot{V}+RU-PW),\] (7) \[Z+mg\cos(\theta)\cos(\phi) = m(\dot{W}+PV-QU), \tag{8}\]  
where glider mass \(m\) (kg), aerodynamic forces \(X\), \(Y\), \(Z\) along body \(x_{b}\), \(y_{b}\), \(z_{b}\)-axis, respectively, axial speeds \(U\), \(V\), \(W\) (m/s), axial acceleration \(\dot{U}\), \(\dot{V}\), \(\dot{W}\) (m/s\({}^{2}\)), and angular speeds \(P\), \(Q\), \(R\) (rad/s). Axial acceleration can be expressed as Equations (9)-(11) and angular speeds can be calculated as Equations (12)-(14) where rolling moment \(L\), pitching moment \(M\), yawing moment \(N\). Axial acceleration and angular speeds are expressed,  
\[\dot{U} = RV-QW-g\sin(\theta)+X/m, \tag{9}\] \[\dot{V} = PW-RU+g\cos(\theta)\sin(\phi)+Y/m,\] (10) \[\dot{W} = QU-PV+g\cos(\theta)\cos(\phi)+Z/m,\] (11) \[\dot{P} = (c_{1}R+c_{2}P)Q+c_{3}L+c_{4}N,\] (12) \[\dot{Q} = c_{5}PR-c_{6}(P^{2}-R^{2})+c_{7}M,\] (13) \[\dot{R} = (c_{8}P-c_{2}R)Q+c_{4}L+c_{9}N, \tag{14}\]  
\begin{table}
\begin{tabular}{l c c} \hline \hline
**Parameters** & **Value** & **Unit** \\ \hline \(V_{t}\) & 8.2 & (m/s) \\ \(V_{c}\) & 6.1 & (m/s) \\ \(V_{max}\) & 12.5 & (m/s) \\ \(C_{L,r}\) & 0.4855 & - \\ \(\alpha\) & 5.5 & (\({}^{\circ}\)) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Determination of the initial necessary parameters for dynamic behavior of the glider.  
where  
\[I=\begin{bmatrix}c_{1}&c_{2}&c_{3}\\ c_{4}&c_{5}&c_{6}\\ c_{7}&c_{8}&c_{9}\end{bmatrix}. \tag{15}\]  
\(Ie\mathbb{R}^{3\times 3}\) represents the moment of inertia of the glider (kgm\({}^{2}\)) [28]. By using Equations (9)-(14), glider stability analysis is done. Solving Equations (9)-(14) yields Euler angle's derivatives in (16)-(18). Euler angles are significant to describe since orientation of a rigid body on the fixed coordinate system could be expressed as  
\[\dot{\phi} =P+\tan(\theta)(Q\sin(\phi)+R\cos(\phi)), \tag{16}\] \[\dot{\phi} =Q\cos(\phi)-R\sin(\phi),\] (17) \[\dot{\psi} =\frac{Q\sin(\phi)+R\cos(\phi)}{\cos(\theta)}, \tag{18}\]  
where \(\dot{\phi}\), \(\dot{\theta}\), \(\dot{\psi}\). By specifying Euler angles, the resulting specific instant speed of the glider can be obtained, which is defined on the body axis. Then, the Equations (19)-(21) could be expressed including axial velocity,  
\[\dot{x} =U\left(c\phi c\theta\right)+V\left(s\phi s\theta c\psi-c\phi s \psi\right)+W\left(c\phi s\theta c\psi+s\phi s\psi\right), \tag{19}\] \[\dot{y} =U\left(s\phi c\theta\right)+V\left(s\phi s\theta s\psi+c\phi c \psi\right)+W\left(c\phi s\theta s\psi-s\phi c\psi\right),\] (20) \[\dot{z} =-U\left(s\theta\right)+V\left(s\phi c\psi\right)+W\left(c\phi c \theta\right), \tag{21}\]  
where \(\cos\) = c, \(\sin\) = s. A method of evaluating the glider's positions on the airframe is to calculate the glide duration.