# Ellipsoidal Path Planning for Unmanned Aerial Vehicles
## 5 Conclusions and Future Work  
In this work we presented a geometrically explicit and simple algorithm that takes advantage of the ellipsoidal representation of maps generated to plan collision-free and smooth paths regardless of the concavity of the obstacles or whether the environment is indoor or outdoor. Our method solves the non-trivial problem of computing the distance between two ellipsoids by using a neural network. To train the neural network, it was necessary to produce a training dataset and design a novel normalization method for these data. So, we obtain an accurate approximation for the distance, which allows the computation of the free-space and occupied-space of an environment, as is shown in Table 2 and Figure 6.  
In order to obtain paths and to keep the computational costs of our approach low, a bio-inspired algorithm named TLBO was used. We have chosen TLBO because it has had the best performance in previous comparisons with other techniques [9]. The fitness function was designed taking into account four desired features for the path obtained: to keep a safe distance between obstacles' ellipsoids and the UAV ellipsoid; to avoid collisions; to reduce the total amount of the UAV's proximity range throughout the whole path; and to give the drone the capability to avoid large convex obstacles (walls inside a room, dense and tall vegetation). \begin{table}
\begin{tabular}{l l l l l l} \hline \hline
**Algorithm** & \multicolumn{1}{c}{**Map 1**} & \multicolumn{1}{c}{**Map 2**} & \multicolumn{1}{c}{**Map 3**} & \multicolumn{1}{c}{**Map 4**} \\ \hline \multirow{2}{*}{TLBO} & mean & 77.1916 & 221.7160 & 98.1027 & 100.3328 \\ \cline{2-6}  & STD & 9.1466 & 145.9610 & 9.8856 & 35.7067 \\ \hline \multirow{2}{*}{DE} & mean & 57.8452 & 110.2755 & 66.7537 & 71.6009 \\ \cline{2-6}  & STD & 10.3801 & 52.5747 & 8.0530 & 19.4613 \\ \hline \multirow{2}{*}{PSO} & mean & 107.8372 & 196.9315 & 111.1351 & 114.3043 \\ \cline{2-6}  & STD & 24.4556 & 15.5697 & 9.4505 & 37.8680 \\ \hline \multirow{2}{*}{FF} & mean & 130.9127 & 257.07965 & 153.2906 & 151.7518 \\ \cline{2-6}  & STD & 19.7207 & 118.1890 & 13.2743 & 30.2673 \\ \hline \multirow{2}{*}{RRT\({}^{\star}\)} & mean & 5.3038 & 2.4780 & 5.8014 & 4.8056 \\ \cline{2-6}  & STD & 0.4246 & 1.0391 & 0.9550 & 0.3053 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Experimental results, mean and standard deviation of the run-time of each algorithm in seconds. Figure 15: Path planning in a real ellipsoidal map, we show the point cloud for a better understanding of the image. The cloud point has 100,000 points but the map only has 700 ellipsoids. It is important to mention that, based on our design of the fitness function, the computed paths are not optimal. Because it was used as a greedy approach, it cannot get the optimal path but is more versatile and can manage dynamical maps without extra computational costs. We compare the proposed approach with other evolutionary algorithms, DE, PSO, and FF. We also compare with RRT*, and we constructed the collision detection function for the ellipsoidal map. In most cases, evolutionary techniques with the presented fitness function obtained a better performance than RRT*, but clearly RRT* has a better performance on time. Furthermore, the evolutionary methods also calculate the UAV orientation and not just the \((x,y,z)\) position. Future WorkThe proposed algorithm uses a greedy strategy to find a nearby optimal position. Therefore, the algorithm is locally optimal. To compare different path planning algorithms, a cost function must be developed, which has non dependency on the map abstraction. We are working on substituting the greedy politic used to compute the path as well as the bio-inspired algorithm by a reinforcement learning algorithm so the system can learn policies of navigation instead of computing paths. Furthermore, we are designing an intelligent low-level control algorithm that considers the dynamical model of the UAV to follow the planned path. Conceptualization, C.V. and J.H.-B. ; methodology, C.V. and A.A.G.