## 2 Materials and Methods
### Problem Formalization and Proposed Method
The measuring activity is done during farming operations. Therefore, we assume the presence of a function \(t\), which returns for each PoI \(p\) the time required to complete the farming operations, including the measuring task. Since the assumed temporal granularity defines a resolution of one minute, the duration \(t(p)\) for each \(p\in\mathcal{P}\) becomes the number of minutes needed to complete the task. * The movement from one PoI \(p_{1}\in\mathcal{P}\) to another PoI \(p_{2}\in\mathcal{P}\) requires some time, which is represented by a function \(m\) that returns for each pair of PoIs the amount of time needed to perform such a movement. Again, since we consider a temporal granularity of one minute, \(m\) returns the number of minutes needed to complete such movement. * The robot has a limited amount of energy charge, which is also consumed by moving among Pols, so it needs to be recharged periodically. For the sake of simplicity, we assume that the available and consumed energy is expressed as an integer representing the battery life in terms of minutes. * Some PoIs contain a charging station in which the robot battery can be charged. The charging activity takes some time to complete, and it does not overlap with farming/measuring operations. For the sake of simplicity, we assume that the charging time depends only on the current capabilities of the charging station, but the model is general enough to include a more complex function that also depends on the charging conditions, like the current battery level. In other words, in practical scenarios, the charging time will depend on the battery level at the moment of recharge, and the robot may require less time to recharge if the battery is not fully depleted. Future work could focus on incorporating a dynamic charging time model based on the robot's actual battery level, enabling more accurate energy management. Integrating variable charging times into the trajectory planning algorithm will improve scheduling efficiency and leads to more realistic and effective operational strategies. In any case, the time required to complete a charging operation is represented by the number of minutes needed to complete it. * The farming/measuring operations are orthogonal to charging operations, and a robot can stop in a PoI to either measure, charge, or do both. An environment characterized by these hypotheses can be formally represented through a graph where the nodes represent Pols, and the edges are the paths connecting them. **Definition 3** (Environment Graph). : _The environment characterizing the problem at hand can be described as a bi-directed labeled graph \(G=(V,E,\mathcal{L},\mathcal{E})\), where:_  
* \(V\) _is the set of nodes representing the Pols._
* \(E\subseteq\{(u,v):u,v\in V,u\
eq v\}\) _is the set of bi-directional edges representing the paths connecting the nodes._
* \(\mathcal{L}:V\rightarrow\mathcal{P}\times\mathbb{N}^{+}\times\mathbb{N}\) _is a labeling function assigning a label_ \(\langle p,t_{p},c_{p}\rangle\) _to each node_ \(u\in V\)_, defined as follows:_
* \(p\in\mathcal{P}\) _is the identifier of the PoI._
* \(t:\mathcal{P}\rightarrow\mathbb{N}^{+}\) _is the function that represents the time required to complete the measuring task together with the farming task inside_ \(p\)_. The notation_ \(t_{p}\) _is used in place of_ \(t(p)\) _for not cluttering the notation,_
* \(c:\mathcal{P}\rightarrow\mathbb{N}\) _indicates whether a given PoI_ \(p\) _contains a charging station, i.e.,_ \(c_{p}=0\) _means that_ \(p\) _does not contain a charging station, while_ \(c_{p}>0\) _represents the time required for the charging operation._
* \(\mathcal{E}:E\rightarrow\mathbb{N}^{+}\times\mathbb{N}^{+}\) _is a function that assigns a label_ \(\langle m_{u,v},e_{u,v}\rangle\) _to each edge_ \(e\in E\) _as follows:_
* \(m:\mathcal{E}\rightarrow\mathbb{N}^{+}\) _is the function that represents the time required by the robot to move between two adjacent PoIs.