## 4 Results on a Real Case: Proof-of-Concept Validation
### Results of the Path Planner System
#### 4.2.4. Path Planning Using Processed Satellite Imagery  
Algorithm 11 is used to calculate the _Speed_ image required to determine the path that the robot should follow. Initially, the _Path binary map_ (Figure 10a) was considered to be used as a base to determine the distances, but a modified version was used instead (Figure 10b). The modification consisted of adding the detected lines for the orange rows. The lines have been added to ensure that  
Figure 8. Extracting reference points: (**a**) determining path row lines perpendicular to tree rows, (**b**) determining bisection between path row lines, (**c**,**d**) intersection between bisection and the Path binary map, and (**e**,**f**) new reference points and its representation on the parcel satellite image. Figure 9. Intermediate points (numbered blue dots) for four different configurations depending on the robot initial position (red circle). The robot must move first to the closest corner (green circle), and then, it must navigate trough the intermediate points (small numbered circles) in order to achieve the last position in the opposite corner (big blue circle). (**a**) robot placed in the upper-left corner, (**b**) robot placed in the bottom-left corner, (**c**) robot placed in the bottom-right corner, (**d**) robot placed in the upper-right corner. the robot navigates entirely through each path row and that there is no way of trespassing the orange tree rows during navigation. Three different _Speed_ images are shown in Figure 11a-c depending on the value of the \(\alpha\) parameter. After performing some experiments, the best results are obtained when the \(\alpha\) value is set to 3. With lower values of \(\alpha\), the points close to obstacles were not penalized. With higher values, some narrow path rows were omitted from navigation due to their low-speed values. Finally, in our experiments, we used \(\alpha=3\), as shown in Algorithm 11. Figure 11a shows the parcel with the destination position drawn on it along with two representations of the _Distances_ matrix (Figure 12b,c). The first one corresponds to the normalized _Distances_ matrix of which the values are adapted to the \([0\dots 255]\) range. However, it is nearly impossible to visually see how to arrive at the destination from any other point inside the parcel. It is due to the high penalty value assigned to those points belonging to obstacles (non-path points). For this reason, a modified way to visualize the distance matrix is introduced by the following equation:  
\[DistImg(x,y)=\left\{ \begin{array}{ll}100+155\frac{Distances(x,y)}{max(Distances )}&\text{if }Obstacles(x,y)=false\\ \\ 0&\text{otherwise}\end{array} \right. \tag{5}\]  
In the image, it can be seen that there are high distance values in the first and last path rows (top-left and bottom-right sides of Figure 12c) which are not present in most of the other path rows. These two rows are narrow, and the speed values are, therefore, also quite low. For example, a robot placed in the top-left corner should navigate through the second path row to achieve the top-right corner instead of navigating through the shortest path (first path row). The aim of Fast Marching is to calculate the fastest path to achieving the destination point, not the shortest one. In the first row, the distance to obstacles is almost always very low. Figure 11: Generating _Speed_ images: (**a**–**c**) graphical representations of _Speed_ values calculated with Equation (4) for \(\alpha\) equal to 1, 2, and 3. Figure 10: Generating _Speed_ images: (**a**) the original _Path binary map_ and (**b**) the modified map. Figure 13 shows the results of the path planner for a robotic system. In the example image, the robotic is initially placed in the location marked with the red x-cross. Then, all the intermediate points, including the closest corner, are determined (green x-crosses). Figure 14 introduces an example for a team composed of two robots. Figure 14a,b shows the complete determined path for the first and second robots.