## 4 Background Information
### Path Planning for Surveillance Tasks  
We consider that the surveillance area is \(S\), and a representation of an environment for MRs that is illustrated in Figure 2. The distribution function of information for the surveillance area is described by \(F(x,\ y)\). The quantity of information is given in Equation (6), for the entire surveillance area is denoted by \(I_{S}\)[36].  
\[I_{S}=\iint\limits_{A_{S}}F(x,y)dxdy, \tag{6}\]  
where \(A_{S}\) is the area of the entire surveillance place. It is typically challenging to gather information from every spot within the surveillance area because of physical constraints. To create the group of remarkable places, we have chosen multiple dispersed locations, and this group is abbreviated as \(R\), which is shown in Equation (7).  
\[\begin{split} R&=\{r_{1},\ r_{2},\ \ldots\ldots \ldots,\ r_{m}\},\\ r_{i}&=i_{(x_{i},\ y_{i})}\end{split} \tag{7}\]  
where the coordinates are \((x_{i},y_{i})\) for the \(i\)th interested place. At such places, an MR conducts inspection operations and obtains information.  
Planner areas can easily be discretized well into basic geometrical simplex sets using Voronoi nets [37]. We split the surveillance region into multiple grids containing obstacles based on the Voronoi concept. Figure 2 provides an instance of the Voronoi segmentation of the surveillance zone within which the MR performs surveillance. The entire surveillance zone is divided into six parts, \(m=6\) and there are total six surveillance locations, which are denoted by \(r_{1},\ r_{2},\ \ldots,\ r_{6}\) respectively.  
Figure 2: Voronoi division of different environments for multiple MR surveillance.  
The mathematical optimization for grids transforms the informational domain of the entire area, and the information related to the places is used for its representation. The quantity of data acquired by the MR is abbreviated in Equation (8) [36]:  
\[I_{M}=\sum_{i=1}^{n}F(x_{i}y_{i})A_{r_{i}} \tag{8}\]  
where \(A_{r_{i}}\) shows the Voronoi space of the surveillance place \(r_{i}\). The number of surveillance places is \(n\), for which real data is gathered. For \(n=m\), the MR will be able to monitor the entire region of the Voronoi diagram and can collect data as well.  
To study how to formulate the issue of a MR's surveillance tasks, we assume that each surveillance place is the coordinate of the path-following MR to perform surveillance tasks. The MR initializes the surveillance task at the first monitored place to gather data and get back to the same place after finishing surveillance work. The distance between each monitored place is given, and the coordinates of each starting place are predefined. The time and energy needed for each surveillance task are also given. The MR velocity and the consumption of energy per unit length are provided.  
Whenever an unusual event is encountered by the MR, the position, the MR's remaining energy, and entry points for the monitored sites cannot be obtainable in the easiest way. The MR must be allowed to ascertain his position now. The proximity concept applies, which means that the accessible acquisition point that is nearest to the MR is considered the beginning place of re-planning, while the autonomous system is traveling between the two places that are considered for the experiment. Acquiring activities are unclear due to variable obstacles within the surroundings. An avoiding obstacle approach needs to be used by an MR while traveling across an unstable obstacle that can choose to stop moving forward, although performing so will delay the work of surveillance. It can additionally accelerate or follow another route, both of which require additional energy. After all criteria are satisfied, an efficient route from the starting position towards every capturing the data position must be identified, which might meet the needs for obtaining the quantity of information possible throughout the surveillance work.