## 2 Kinetic Launcher Idea Description and Mathematical Model
### Mathematical Description  
In the system described in the previous subsection we introduced the angular speed of the flywheel \(\omega_{s}(t)\) and the angular speed of the driving gears. These two variables are connected through the torque transmitted by the active electromagnetic clutch \(M_{t}(t)\). Thus, the dynamic equation involving viscous losses in the flywheel bearings can be expressed in the following way:  
\[I_{s}\frac{d\omega_{s}\left(t\right)}{dt}=-M_{t}\left(t\right)-b_{s}\omega_{s }\left(t\right) \tag{1}\]  
The dynamic equation of the driving mechanism is more complex. It involves the torque transmitted by the active electromagnetic clutch \(M_{t}(t)\), the braking torque caused by the active electromagnetic clutch \(M_{h}(t)\), viscous losses in the bearings of rotating parts (\(b_{n}\omega\)), viscous losses in the linear track (\(b_{nv}\)), aerodynamic drag (\(b_{areco}\)), gravity (\(g\)) and Coulomb friction (\(\mu_{n}\)). The impact of the above mentioned parameters has been expressed using the following formula:  
\[I_{n}\frac{d\omega_{n}\left(t\right)}{dt} =M_{t}\left(t\right)-M_{h}\left(t\right)-b_{n\omega}\omega_{n} \left(t\right)-b_{nv}{r_{n}}^{2}\omega_{n}\left(t\right)-b_{areo}{r_{n}}^{2} \omega_{n}\left(t\right)f_{sep}\left(t\right)-\] \[\left({r_{n}}^{2}\frac{d\omega_{n}\left(t\right)}{dt}+r_{n}g\sin \left(\alpha\right)+\mu_{n}g{r_{n}}\cos\left(\alpha\right)sign\left({r_{n} \omega_{n}\left(t\right)}\right)\right)\left({m_{w}+m_{o}f_{sep}\left(t\right) }\right).\]  
In order to describe how the UAV disengages from the launcher, we have introduced a launching carriage/UAV separation function \(f_{sep}(t)\) which can be expressed in the following manner:  
\[f_{sep}\left(t\right)=\left\{\begin{array}{ll}1&\text{for}\quad x\left(t \right)\leq x_{h}\\ 0&\text{for}\quad x\left(t\right)>x_{h}\end{array}\right.. \tag{2}\]  
Parameter \(x_{h}\) represents the linear distance where the braking phase starts. Assuming a lack of flexibility of the transmission toothed belt we can create equations for linear displacement \(x(t)\), velocity \(v(t)\) and acceleration \(a(t)\) of the launcher carriage with the attached UAV [23].  
\[x\left(t\right)=r_{n}\int\omega_{n}\left(t\right)dt. \tag{3}\]  
\[v\left(t\right)=r_{n}\omega_{n}\left(t\right). \tag{4}\]  
\[a\left(t\right)=r_{n}\frac{d\omega_{n}\left(t\right)}{dt}. \tag{5}\]The dynamics of the active electromagnetic clutch and the brake can simply be described as a standard \(RL\) circuit [23; 24]. Equations involving time constants \(T_{t}\), \(T_{h}\) and maximum torque capacities \(M_{t0}\), \(M_{h0}\) are presented below:  
\[T_{t}\frac{dM_{t}\left(t\right)}{dt}+M_{t}\left(t\right)=M_{t0}u_{t}\left(t \right). \tag{6}\]  
\[T_{h}\frac{dM_{h}\left(t\right)}{dt}+M_{h}\left(t\right)=M_{h0}u_{h}\left(t \right). \tag{7}\]  
Transmitted and braking torques can be controlled, respectively, by binary \(u_{t}(t)\) and \(u_{h}(t)\) signals. They can be generated as the output of the control algorithm with feedback implemented on a microcontroller. One of the simplest approaches to this is to use the \(x_{h}\) parameter.  
\[u_{t}\left(t\right)=\left\{\begin{array}{ll}1&\text{for}\quad x\left(t \right)\leq x_{h}\\ 0&\text{for}\quad x\left(t\right)>x_{h}\end{array}\right.. \tag{8}\]  
\[u_{h}\left(t\right)=\left\{\begin{array}{ll}0&\text{for}\quad x\left(t \right)\leq x_{h}\\ 1&\text{for}\quad x\left(t\right)>x_{h}\end{array}\right.. \tag{9}\]