# A High-Fidelity Energy Efficient Path Planner for Unmanned Airships
## 5 Trajectory Objectives
### Energy Consumption  
The energy objective cost function, \(\mathbb{C}_{e}\), can be defined by the energy required to traverse two nodes. Considering the vehicle described in Section 3, \(\mathbb{C}_{e}\) is given by,  
\[\mathbb{C}_{e}\left(n,n^{\prime}\right)=\frac{\left(P_{c}+P_{L}+P_{T}\right) \cdot t\left(n,n^{\prime}\right)}{E_{c}} \tag{6}\]  
where \(P_{T}\) is the power required to propel the vehicle at the desired ground speed along the horizontal path, \(P_{L}\) is the power required to change altitude and \(P_{c}\) is the constant electrical power required to maintain on-board electronics (e.g., camera, flight controller, telemetry, etc.). The inclusion of \(P_{c}\) is significant as it can represent up to 65% of the total power for airships [48]. The addition of this parameter also prevents unrealistic loitering when a large time goal is set in the planner. The platform simulated in this paper has constant on-board electronic power requirements of \(P_{c}=50\) W [38]. The energy constant \(E_{c}\) is used to normalize the energy cost. It is set at \(v_{vw}=v_{v,max}\) with zero net wind (computed to be 1.50 Wh for the proposed platform). Substituting the definitions of mechanical power for propeller and pump actuators into (6) gives,\[\mathbb{C}_{e}\left(n,n^{\prime}\right)=\left(P_{c}+\frac{Q\triangle\rho\left(n,n^{ \prime}\right)}{\eta_{pump}}+\frac{F_{T}}{\eta_{T}}\left|v_{vw}\right|\right) \cdot\frac{t\left(n,n^{\prime}\right)}{E_{c}} \tag{7}\]  
where \(-12\) L/s \(<Q<12\) L/S is the air flow rate through the air ballast pump for the given platform [40], \(\triangle\rho\left(n,n^{\prime}\right)\) is the change in atmospheric pressure between nodes and \(F_{T}\) is the force required from the UAV's thrusters. The propeller forces applied by the UAV thrusters, including the electric motors and the hybrid power plant, are non-linear functions of input electrical power. A propeller's efficiency, \(\eta_{T}\), is dependent on the propeller advance ratio, which is the ratio of the vehicle's relative velocity \(v_{vw}\) over the product of the propeller's diameter and rotational speed. Based on a third order least squares fit of propeller efficiency curves for the 12-inch propeller used by the simulated platform, the propeller efficiency can be estimated from,  
\[\eta_{T}\approx-2.59\cdot 10^{-5}v_{vw}^{3}+2.07\cdot 10^{-4}v_{vw}^{2}+4.15 \cdot 10^{-2}v_{vw} \tag{8}\]  
for \(0<v_{vw}<36\) m/s [38]. The wind vector field is a crucial addition to the energy consumption model as longitudinal winds will effect the arrival time in (3) and compounds the effect of applied lateral winds in (7). For a force equilibrium,  
\[F_{T}=F_{d}=\frac{1}{2}C_{d}\delta_{air}A\left|v_{vw}\right|^{2} \tag{9}\]  
The product of the axial drag coefficient and reference area, \(C_{d}A\), in (9) can be estimated by a few methods. Using the empirical method proposed by [PERSON] [49], the value for this platform is 0.229. Using the procedure outlined by [PERSON] [50], the value is 0.147. Furthermore, the value is not constant for all flight conditions, and increases with both the angle of attack and the angle sideslip. Therefore, for the airship to hold a constant crab angle into the wind is not trivial from the perspective of energy consumption. In this paper, the value predicted by [PERSON] was used as it was developed explicitly for airships [50]. The air ballast flow rate, \(Q\), is assumed to be equal to the ballast volume change, \(\triangle V_{b}\), required to maintain vehicle buoyancy with changes in altitude over a given travel time,  
\[Q=\frac{\triangle V_{b}\left(n,n^{\prime}\right)}{t\left(n,n^{\prime}\right)} =\frac{\triangle\delta_{air}\left(n,n^{\prime}\right)V_{v}}{\delta_{air} \left(n\right)t\left(n,n^{\prime}\right)} \tag{10}\]  
where \(V_{v}\) is the envelope volume of vehicle. Substituting (3), (9) and (10) into (7) yields the complete energy objective function,  
\[