# Design and Implementation of a Tether-Powered Hexacopter for Long Endurance Missions
## 2 Design and Analysis
### Tether Optimization  
The maximum range or altitude of a tether-powered multicopter is limited by the tether length and the power that can be used. Considering the condition where the hexacopter hovers right above the ground station, the length of tether is equal to the altitude. The power needed by the hexacopter, \(P_{n}\), for hovering in the air can be expressed as Equation (1). \[P_{n}=\frac{1}{\eta}\left[n\left[f\left(\frac{W_{IA}}{n}\right)\right]+P_{AV}+ P_{PL}\right] \tag{1}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Specifications** & **Description** \\ \hline Weight of airframe & 2000 g \\ Hub-to-hub (diagonal) dimension & 680 mm \\ Flight controller (avionics) & 3 DR Pixhawk with ArduPilot \\ Electrical speed controller (ESC) & Hobbywing Platinum 30A Pro 2-65 ESC \\ Brushless DC Motor & SunnySky V3508-29 KV380 \\ Propeller & 1255 carbon fiber propeller \\ Telemetry & 915 MHz transmitter \\ Power consumption by Avionics & 10 W approximately \\ \hline Weight of power supply module (PSM) & 696 g \\ Nominal output voltage of PSM & 24 V \\ Nominal output power of PSM & 600 W \\ Energy capacity of backup battery & 36 Wh \\ \hline Payload capacity & 1500 g \\ Weight of mission payload & 170 g \\ Power consumption by mission payload & Less than 5 W \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Specifications of the proposed hexacopter subsystem. Figure 2: (**a**) The power supply module (PSM) consists of a DC-DC convertor, a heat sink, cooling fans, and a backup battery for emergency landing; (**b**) The thermal image of the DC-DC convertor providing 30 A at 24 V.  
where \(\eta\) represents the efficiency of the DC-DC convertor of PSM and is fixed to 0.9307 as the worst case. The symbol \(n\), which is 6 for a hexacopter, represents the quantity of rotors. The function \(f(\prime)\) represents the thrust-to-power function of a single rotor where the input unit is gram-force and the output unit is Watt. \(W_{IA}\) represents the weight in the air. \(P_{AV}\) and \(P_{PL}\) represent the power consumed by avionics and the payload, respectively. The thrust-to-power function used in this research work is acquired from our test bed described in [18]. The curve fitting with the momentum theory [25] can be described by Equation (2). \[P_{n}=\frac{F_{r}^{\frac{3}{2}}}{\eta_{r}\sqrt{2\rho_{a}A_{r}}} \tag{2}\]  
where \(F_{r}\) is the thrust force produced by the rotor, \(\eta_{r}\) is a figure of merit of the rotor, \(\rho_{a}\) is the density of air, and \(A_{r}\) is the swept area of the propeller. The test data are shown in Figure 3, and the \(3/2\) order fitted regression becomes Equation (3). \[f(F_{r})=0.0039F_{r}^{\frac{3}{2}}+1.95 \tag{3}\]  
A none zero constant 1.95 in Equation (3) implies the static power loss of ESC when the rotor is not rotating.\(W_{IA}\) can be represented as  
\[W_{IA}=W_{TO}+dH \tag{4}\]  
where \(W_{TO}\) is the takeoff weight of the hexacopter, \(d\) is the tether's weight of per unit length, \(H\) is the length of the tether, and \(dH\) is the weight of tether in the air. The power submitted by the ground station, \(P_{G}\), can be expressed as  
\[P_{G}=V_{G}I_{G}=I_{G}^{2}\rho H+P_{r} \tag{5}\]  
where \(V_{G}\) and \(I_{G}\) represent voltage and current submitted by the ground station respectively. The symbol \(\rho\) is the tether's resistivity per unit length thus the power consumed by the tether can be represented as \(I_{G}^{2}\rho H\). \(P_{r}\) is the residual power that the hexacopter can receive. The maximum power that can be transferred from the ground station to the hexacopter is limited by the minimum allowable input voltage of the DC-DC converter (\(V_{DCH,MIN}\)). For most DC-DC converters, an under voltage trip is likely to occur if input voltage (\(V_{DCH}\)) drops below \(V_{DCH,MIN}\). This limitation can be expressed as  
\[V_{G}-I_{H}\rho H\geq V_{DCH,MIN} \tag{6}\]  
Figure 3: The thrust-to-power function of a single rotor.