# Mini Glider Design and Implementation with Wing-Folding Mechanism
## 2 Mathematical Model
### Aerodynamic Calculations  
The aerodynamic force coefficients need to be determined in order to perform necessary calculations when the glider is hovering in a helical orbit. To determine these coefficients, the Vortex lattice method, (VLM), is used which is a numerical method used in computational fluid dynamics. For this purpose, XFLR5\({}^{\textregistered}\) aerodynamic analysis is used to determine the position of the glider on the \(x\)-, \(y\)-, and \(z\)-axes after the instant speed of the glider is obtained in Section 2.2 To determine position of the glider, stability and control coefficients have to be calculated. Equation of coefficients are shown in (22)-(29):  
Figure 3: The center of gravity of the glider is indicated by “O” and it is given by the front view. The parameters in Equations (6)–(8) are expressed according to earth-fixed frame and body-fixed frame. Using the relationship of these frames, nonlinear equations can be derived.  
\[c_{L} = c_{L,0}+c_{L,\alpha}, \tag{22}\] \[c_{D} = c_{D,0}+kc_{L}^{2},\] (23) \[c_{x} = -c_{D}c(\alpha)+c_{L}s(\alpha),\] (24) \[c_{y} = -c_{y,\beta}\beta+c_{y,p}P(\frac{b}{2V})+c_{y,p}R(\frac{b}{2V})+c_ {y,\alpha_{\delta}}\delta_{a}+c_{y,\delta}\delta_{r},\] (25) \[c_{z} = -c_{L}c(\alpha)-c_{D}s(\alpha)-c_{L,q}Q(\frac{c}{2V})-c_{L,\delta _{r}}\delta_{e},\] (26) \[c_{l} = c_{l,\beta}\beta+c_{l,p}P(\frac{b}{2V})+c_{l,r}R(\frac{b}{2V})+c _{l,\delta_{r}}\delta_{a}+c_{l,\delta_{r}}\delta_{r},\] (27) \[c_{m} = c_{m,0}+c_{m,\alpha}\alpha+c_{m,\delta}Q(\frac{c}{2V})+c_{m, \delta_{r}}\delta_{e},\] (28) \[c_{n} = c_{n,\beta}\beta+c_{n,p}P(\frac{b}{2V})+c_{n,r}R(\frac{b}{2V})+c _{n,\delta_{a}}\delta_{a}+c_{n,\delta_{r}}\delta_{r}, \tag{29}\]  
nonlinearly with flow angles (\(\alpha,\beta\)), control surface deflections (\(\delta_{e},\delta_{a},\delta_{r}\)), chord length \(c\), wingspan \(b\), \(x\) direction \(c_{x}\), \(y\) direction \(c_{y}\), \(z\) direction \(c_{z}\), rolling moment \(c_{l}\), pitching moment \(c_{m}\), yawing moment \(c_{n}\)[27]. The obtained coefficients are used to calculate \(X,Y,Z,L,M,N\)[29]. Therefore, coefficents, which will be used for aerodynamic forces and moments acting on glider, are shown in Table 3.  
According to these coefficients, axial speeds and moments can be expanded into a set of six equations that could be used to determine \(X,Y,Z,L,M,N\) in Equations (30)-(35):  
\[X=c_{x}qS, \tag{30}\] \[Y=c_{y}qS,\] (31) \[Z=c_{z}qS,\] (32) \[L=c_{l}qS,\] (33) \[M=c_{m}qSc,\] (34) \[N=c_{n}qSb, \tag{35}\]  
where dynamic pressure constant is \(q\) and wing area is \(S\). The resultant model with the equipment is shown in Figure 4. In this figure, the communication & GPS unit provides the glider position, the camera unit is responsible for capturing images when the glider is launched (inside of the rocket), after first separation (inside of container) and after separation of the container. Lastly, the pitot tube and pressure sensor measure appropriate flight data.  
Sensors collect the necessary information, and the data are recorded on an Secure Digital (SD) card and also transmitted to the ground station during glide. The battery is used to provide the necessary energy. Equipment is modelled especially so that the center of gravity is placed at the wing's leading edge (1/3) in the glider.  
\begin{table}
\begin{tabular}{l c} \hline \hline
**Longitudinal** & **Lateral** \\ \hline \(c_{L,\alpha}=4.8878\) & \(c_{y,\beta}=-0.36561\) \\ \(c_{L,q}=8.6232\) & \(c_{n,\beta}=0.09735\) \\ \(c_{D,0}=0.07\) & \(c_{l,\beta}=-0.17966\) \\ \(c_{D}=0.1065\) & \(c_{l,\beta}=-0.17966\) \\ \(c_{m,\alpha}=-3.0688\) & \(c_{y,\
u}=0.3857\) \\ \(c_{m,q}=-20.781\) & \(c_{n,r}=-0.12476\) \\  & \(c_{l,r}=-0.22072\) \\  & \(c_{y,p}=-0.30632\) \\  & \(c_{n,p}=-0.10115\) \\  & \(c_{l,p}=-0.58491\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Aerodynamic force coefficients.