## 2 Kinetic Launcher Idea Description and Mathematical Model
### Structure of Kinetic Launcher  
Figure 3 shows the parts of the kinetic launcher's rotating drive and presents some crucial signals and parameters.  
* - flywheel damping rate, [Nm/rad/s];
* - initial angular speed of the flywheel, [rad/s];
* - flywheel moment of inertia, [kgm\({}^{2}\)];
* - angular speed of the flywheel, [rad/s];
* - driving torque transmitted by the clutch, [Nm];
* - electromagnetic clutch control signal, [dimensionless];
* - damping rate of the rotating drive elements, [Nm/rad/s];
* - radius of the drive wheels of the launcher, [m];
* - total moment of inertia of the driving and driven wheels, [kgm\({}^{2}\)];
* - angular speed of the driving tooth wheels, [rad/s];
* - braking torque transmitted by the clutch, [Nm];
* - electromagnetic brake control signal, [dimensionless].  
Figure 2: The rotating drive part of the kinetic launcher.  
The damping of the rotational motion of the flywheel and the drive were considered in the model. The DC motor is necessary to put the flywheel into motion and bring it up to an appropriate speed. After that, the motor is idle and rotates freely together with the flywheel. In the construction of the device three shafts were proposed: the flywheel shaft, the driving shaft and the braking shaft. All shafts are mounted on bearings. Torque transmission between the shafts is achieved through an active electromagnetic clutch and a brake controlled using dimensionless u\({}_{t}\)(t) and u\({}_{h}\)(t) signals, respectively. On the side opposite the flywheel, the braking shaft is fixed to the launcher frame.  
Figure 4 depicts the synchronous drive transmission subsystem.  
* - gravitational acceleration, [m/s\({}^{2}\)];
* - Coulomb friction coefficient, [dimensionless];
* - aerodynamic damping rate, [N/m/s];
* - mass of UAV, [kg];
* - mass of launcher cart, [kg];
* - velocity of the UAV and launcher cart, [m/s];
* - damping rate of the linear motion, [N/m/s];
* - angle of launch, [rad];
* - angular speed of the driven wheels, [rad/s].  
Figure 4: The synchronous drive transmission subsystem.  
Figure 3: Parts of the kinetic launcher’s rotating drive.  
In order to transmit motion from the driving gears to the driven gears we proposed using a pair of synchronous toothed belts. This is a well known textbook approach often used in these types of solutions [23]. The toothed belts have to display some level of resilience and elasticity since we are aware that they can significantly influence the behavior of the device. We have introduced the \(\omega_{n}(t)\) and \(\omega_{n}^{\prime}(t)\) signals which describe the angular speed of the launcher's driving and driven gears, respectively. Nevertheless, if we take into account the relatively not-so-small total length and negligibly low flexibility of the belts, we can simply assume that: \(\omega_{n}(t)=\omega_{n}^{\prime}(t)\). Thus, we are certain that \(v(t)=r_{n}\omega_{n}(t)\) holds over the entire distance required to launch of UAV. In the model, we introduced two masses. One of them is connected with the launching carriage and other linearly moving parts while the second represents the UAV being launched. These masses strongly impact the dynamic behaviors of the system both in the starting as well as in the braking phase.