# A High-Fidelity Energy Efficient Path Planner for Unmanned Airships
## 6 Numerical Simulations
### Results and Discussion  
Figure 8 shows a typical environment with the planned trajectory plotted for each planner and cross-sections of the variable WVF. In this example, the start and goal nodes are set to \([0.5,1,0.25]\) and \([1.75,1.75,0.5]\) km, respectively. The wind seed, represented by a light blue line, is approximately located at \([2,0.5,0.75]\) km, has a \([-0.707,0.707,0]\) direction, and a magnitude of 3.25 m/s. The darker regions of the WVF cross-sections in the figure indicate high wind magnitudes. The WVF is computed at the start of the planning process and remains constant during the simulation. The performance of the proposed complete planner was bench-marked against that of the other three. The aggregated results of 100 simulations are summarized in Table 2. It is important to note that the average computation time, the average flight time and the average energy consumption were only computed for instances where all four cases had produced a feasible trajectory. It is also important to note that buildings are randomly generated in each new simulation. Generally speaking, however, the wind amplitude and turbulence is much larger around buildings than away from buildings, as will be discussed at the end of this section. Therefore, when combined with the MSZ, the airship tends to plan paths well away from buildings. Figure 8: A typical planned trajectory in a simulated city landscape showing WVF magnitude cross-sections. Note: please refer to the full-color PDF. The first comparison, shown in Table 2a, illustrates the impact of using a low terrain resolution planner. The high resolution trajectory planner converged to a viable solution 94/100 times as opposed to 57/100 for the low resolution version. The convergence rate for the shortest straight line case is understandably low since the vehicle has limited on-board power and the number of passable nodes is limited. Furthermore, the average energy computed in the low resolution case for the trajectories that did converge was over four-times higher than the high resolution version. The lower computation time for the low terrain resolution planner was caused by many of the potential trajectory options being eliminated when optimizing for the shortest distance instead of minimum time and/or energy. However, the computation time for both planners was still only a fraction of the predicted flight time. The second comparison, shown in Table 2b, illustrates the impact of using a temporal vehicle velocity. The paths generated without velocity optimization were generally spatially similar to the complete trajectory planner. However, the constant velocity planner exhibited a lower convergence due its inability to overcome stronger lateral and head winds. Excluding velocity in the path planning process also excludes the option of planning the time the UAV will be in a given wind field. To fully optimize energy consumption, it is desirable to remain in tail winds for a longer period of time (by reducing airspeed) and passing through strong lateral and head winds as quickly as possible. To further illustrate this effect, a case study was conducted to highlight the function selecting the vehicle velocity to minimize the energy consumption defined in (11). Consider the vehicle flying horizontally over a straight line distance of 100 m, with varying amounts of parallel and perpendicular wind components, \(-10\,\mathrm{m}/\mathrm{s}\leq v_{vw,||}\leq 5\,\mathrm{m}/\mathrm{s}\) and \(v_{vw,\perp}=0\), 3, 6 m/s, respectively. If these wind components are known, the vehicle velocity, \(v_{vw}\), can be chosen to minimize energy, as shown by the white line on each surface in Figure 9. Clearly, since the line in each graph is nonlinear, the desired vehicle velocity for minimal energy consumption is not constant and should be included in the planning process. Furthermore, Figure 9 shows that the optimal vehicle velocity is never zero even in the presence of a pure tail wind due to the non-negligible constant power required to maintain the on-board electronics.