# Enhanced Path Planning by Repositioning the Starting Point +
## 3 Proposed Method
### Adjusting the Starting Point  
Thus far, it appears that the starting point needs to be adjusted to a better position that serves as the optimal position for a drone to travel to its connecting stations in the shortest possible distance. To achieve this, the following approaches were chosen in order to determine the best position for the task at hand. Figure 5: SPF factor (0.7) close to the maximum value of 1. Center of the field. This is the point where the diagonals of the area under investigation cross (see Figure 6), and is thus selected irrespective of the stations' locations. It is defined in Equation (5) as  
\[\text{StartingPoint}_{field}=\left(\frac{x_{min}+x_{max}}{2},\frac{y_{min}+y_{ max}}{2}\right) \tag{5}\]  
where \((x_{min},y_{min})\) and \((x_{max},y_{max})\) are the coordinates of the field boundaries. Center of minimum bounding rectangle (MBR) [46]. The point where the diagonals of the MBR intersect (see Figure 7). It is defined in Equation (6) as  
\[\text{StartingPoint}_{MBR}=\left(\frac{x_{min}^{all}+x_{max}^{all}}{2},\frac{y _{min}^{all}+y_{max}^{all}}{2}\right) \tag{6}\]  
where \((x_{min}^{all},y_{min}^{all})\) and \((x_{max}^{all},y_{max}^{all})\) are the coordinates of the minimum bounding rectangle containing all stations. Centroid of all stations [47,48]. The \(x\) and \(y\) coordinates of the starting points are obtained by averaging the \(x\) and the \(y\) coordinates of the \(n\) stations in the field (see Figure 8). It is defined in Equation (7) as  
\[\text{StartingPoint}_{centroid}=\left(\frac{\sum_{i=1}^{n}x_{i}}{n},\frac{\sum _{i=1}^{n}y_{i}}{n}\right) \tag{7}\]  
where \((x_{i},y_{i})\) are the coordinates of the \(n\) stations in the field. Figure 6: Center of the field. Figure 7: Center of MBR. MBR in SPD. Similar to approach (b), but this time focusing exclusively on the starting and ending stations (endpoints) of each route (see Figure 9). It is defined in Equation (8) as  
\[\text{StartingPoint}_{MBR\_SPD}=\left(\frac{x_{min}^{SPD}+x_{max}^{SPD}}{2}, \frac{y_{min}^{SPD}+y_{max}^{SPD}}{2}\right) \tag{8}\]  
where \((x_{min}^{SPD},y_{min}^{SPD})\) and \((x_{max}^{SPD},y_{max}^{SPD})\) are the coordinates of the minimum bounding rectangle containing only the endpoints of each route. Centroid in SPD. Similar to approach (c), but this time focusing exclusively on the endpoints of each route (see Figure 10). It is defined in Equation (9) as  
\[\text{StartingPoint}_{centroid\_SPD}=\left(\frac{\sum_{j=1}^{m}(x_{j,first}+x_{ j,last})}{2m},\frac{\sum_{j=1}^{m}(y_{j,first}+y_{j,last})}{2m}\right) \tag{9}\]  
where \((x_{j,first},y_{j,first})\) and \((x_{j,last},y_{j,last})\) are the coordinates of the first and last stations of route \(j\), \(m\) is the number of routes, and \(2m\) is the number of points (each route contributes \(2\) points: first station and last station). Figure 8: Centroid of all stations. Figure 9: MBR in SPD. * Geometric median in SPD [49]. The point \(y\) that minimizes the sum of distances to all endpoint coordinates of each route (see Figure 11). It is defined in Equation (10) as \[\text{StartingPoint}_{field}=\left(\frac{x_{min}+x_{max}}{2},\frac{y_{min}+y_{ max}}{2}\right) \tag{5}\]0 (10) where \[\text{StartingPoint}_{field}=\left(\frac{x_{min}+x_{max}}{2},\frac{y_{min}+y_{ max}}{2}\right) \tag{5}\]1 (g) Brute force in SPD. Test every point in the MBR area formed by the endpoints of each route and see which one is the optimal point to act as the starting point (see Figure 12). It is defined in Equation (11) as  
\[\text{StartingPoint}_{field}=\left(\frac{x_{min}+x_{max}}{2},\frac{y_{min}+y_{ max}}{2}\right) \tag{5}\]2 Geometric median in SPD. \[\text{StartingPoint}_{field}=\left(\frac{x_{min}+x_{max}}{2},\frac{y_{min}+y_{ max}}{2}\right) \tag{5}\]3 Centroid in SPD. \[ \begin{split}\text{StartingPoint}_{brute\_force}=\underset{(x,y)\in \text{Grid}_{SPD}}{\operatorname*{arg\,min}}\sum_{j=1}^{m}\Bigg{(}& \sqrt{(x-x_{j,first})^{2}+(y-y_{j,first})^{2}}\\ &+\sqrt{(x-x_{j,last})^{2}+(y-y_{j,last})^{2}}\Bigg{)}\end{split}