## 4 Optimal Motion Planning Using Voronoi Partitions and Prediction Difference Maps  
The GPR predicted full-map provides the basis for the optimal motion planning procedures at every instance a prediction is made. We adopt the method originally described by [PERSON] et. al. that utilizes Voronoi diagrams as a means to achieve ideal coverage for distributed networked mobile sensors [12; 13; 14]. Some of the key advantages of using Voronoi partitions to establish optimal motion planning include (a) the method inherently includes UAV-sensor collision avoidance, which can be a complicated problem to solve [27], and (b) the method is scalable to any number of UAV-sensors [12; 13; 14]. Given discrete grid locations \(g\) within a predefined area \(G\subset R^{2}\), and a set \(S=\{s_{1},...,s_{N}\}\subset G\) of \(N\) sensors in distinct locations where \(s_{n}\) contains the x and y coordinates, we define the Voronoi partitions generated by \(S\) as the collection of sets \(V=V_{1}(S),\,\ldots\,\,V_{N}\) (\(S\)) where \(V_{i}(S)\) is defined as:  
\[V_{n}(S)=\{g\in G\mid\ \|g-s_{n}\|\leq\|g-s_{m}\|,\ \ \ for\ all\ s_{m}\in S\} \tag{11}\]  
and refer to (11) as the Voronoi cell of sensor \(s_{n}\)[12; 13; 14; 16; 17; 18; 19]. Equation (11) states that the Voronoi cell \(V_{n}\) consists of all the points within the entire area of interest whose distance to sensor \(s_{n}\) is less than or equal to its distance to any other sensor \(s_{m}\). Suppose we have a recursive framework where \(j\) represents each instance of observations from \(n\) sensors. For every iteration \(j\), the GPR model predicts the full-map \(\phi_{j}(g)\) and the prediction-difference map (PDM), defined as the absolute difference between the current and prior GPR predicted full-map is calculated by using  
\[\Psi(g)=\big{|}\phi_{j}(g)-\phi_{j-1}\left(g\right)\big{|}. \tag{12}\]  
The PDM is used as the density function describing the difference in prior and current predicted features throughout the entire area. The full PDM is partitioned for each Voronoi cell (i.e., each sensor region), \(\Psi_{n}\), which describes a density function outlining the largest discrepancies between successive observations. Using the concepts of centroidal Voronoi tessellations, it was demonstrated in [13] that the local minimum points for optimizing the objective function in each Voronoi cell are simply the density-function-weighted centroids. Therefore, we establish a navigation policy that is based on moving toward the direction of the center of mass by using the PDM as the spatial density function to describe the contamination in each Voronoi cell. The mass and the center of mass for PDM-weighted Voronoi cell are calculated by using  
\[M_{\Psi_{n,\ j}}=\sum_{V_{n,j}}\Psi_{n} \tag{13}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Kernel**} & **Radial Basis** & **Rational** & **Mätérn** & **Mätérn** & **Mätérn** \\  & **Function** & **Quadratic** & **(\(\
u=\frac{1}{2}\))** & **(\(\
u=\frac{3}{2}\))** & **(\(\
u=\frac{5}{2}\))** \\ \hline Functional Form & \multirow{2}{*}{\(\exp\left(-\frac{r^{2}}{2^{2}}\right)\)} & \multirow{2}{*}{\(\exp\left(1+\frac{r^{2}}{2^{2}}\right)^{a}\)} & \multirow{2}{*}{\(\exp\left(-\frac{1}{1}\cdot r\right)\)} & \multirow{2}{*}{\(\begin{bmatrix}1+\frac{\sqrt{3}}{l}\cdot r\\ \exp\left(-\frac{\sqrt{3}}{l}\cdot r\right)\end{bmatrix}\)} & \multirow{2}{*}{\(\begin{bmatrix}1+\frac{\sqrt{5}}{l}\cdot r\\ \exp\left(-\frac{\sqrt{5}}{l}\cdot r\right)\end{bmatrix}\)} \\ R(r) & & & & \\ \hline Fold 1 & 0.118 & 0.057 & 0.052 & 0.060 & 0.069 \\ \hline Fold 2 & 0.133 & 0.057 & 0.051 & 0.061 & 0.071 \\ \hline Fold 3 & 0.131 & 0.053 & 0.048 & 0.060 & 0.070 \\ \hline Fold 4 & 0.121 & 0.056 & 0.051 & 0.062 & 0.072 \\ \hline Fold 5 & 0.141 & 0.055 & 0.052 & 0.062 & 0.074 \\ \hline
**Average** & **0.129** & **0.056** & **0.051** & **0.061** & **0.071** \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Five-fold cross-validation results for all tested covariance functions.