## 5 Proposed Methodology
### Mathematical Analysis and Proof  
We used the methodologies mentioned in [36] to formulate this experiment for the simulation. Let us consider that \(S\) is the collection of routes linking the two surveillance places. \(S=\left\{\,l_{ij}|i\
eq j,i,j\in R\right\}\), where \(l_{ij}\) shows the path length between \(r_{i}\) and \(r_{j}\) for MR. It is possible that the MR might move across dynamic obstacles while it accomplishes the surveillance objectives between the two gathering spots. The probability that an MR will come across an obstacle along its intended path has been designated by a set \(P\). \(P=\left\{P_{l_{ij}}^{*}\right|i\
eq j\lor i,j\in R\right\}\), where \(P_{l_{ij}}^{*}\) denotes the probability of facing obstacles by an MR on the route \(l_{i,j}\). The MRs need more energy while facing obstacles and delays of the surveillance problems; thus, it is required during the planning of the path to encounter fewer obstacles while navigating. The interoperability of the surveillance places seems to be taken for assumption. For the real project, whenever a path is not available between \(r_{i}\) and \(r_{j}\), we can fix \(l_{i,j}=\infty\) and \(P_{l_{ij}}^{*}=1\). Let the series of surveillance places dependent upon a specific route of the MR passing along the surveillance zone as \(R^{n}\). Where \(R^{n}\subseteq R\). \(Q^{n}\) is assigned as the collection in route lengths generated by \(R\). Where \(Q^{n}\subseteq S\). MRs traveling on such paths have a sequence of confronting obstacle probabilities that can be expressed as \(P^{n}\). Where \(P^{n}\subseteq P\). The quantity of information gathered by the MR from the surveillance zone is estimated by Equation (9) [36]:  
\[I_{M}\big{(}R^{n}_{i}Q^{n}_{i}P^{n}\big{)}=\sum_{i=1,r_{i}\in R^{n}}^{n}F(x_{i} y_{i})A_{r_{i}} \tag{9}\]  
where \(A_{r_{i}}\) shows the Voronoi region of surveillance location \(r_{i}\). The total number of surveillance places is \(n\), for the MR where actual surveillance happened. Whenever the MR gathers the entire data from the desired sites, the \(m=n\) and \(R^{n}=R\). The complete length of every planned route for MR is abbreviated as \(dist_{M}\) in Equation (10) [36]. \[dist_{M}\big{(}R^{n}_{i}Q^{n}_{i}P^{n}\big{)}=\sum_{l_{ij}\in Q^{n}}l_{i,j} \tag{10}\]  
The route has been designed to ensure it faces a minimal likelihood of striking obstacles with the objective of lowering a MR's energy use and shortening the duration required to acquire information. For an MR, the median possibility of hitting obstacles throughout each section of a predefined route might be estimated using Equation (11) [36]. \[obst_{M}\big{(}R^{n}_{i}Q^{n}_{i}P^{n}\big{)}=\frac{\sum_{l_{ij}\in P^{n}}^{n} P_{l_{ij}}^{n}}{n-1} \tag{11}\]  
We suppose that while performing surveillance mission, the MR will move with a constant velocity of \(v_{n}\). The energy required for the MR per unit length is \(E_{I}\). The required energy for the MR to carry out the surveillance work at the surveillance location is \(E_{r}\). The time required for the MR to perform surveillance is \(t_{r}\). The initial energy of MR before starting the surveillance task is \(E_{initial}\). For accomplishing the surveillance task, we allocated a time of \(t_{initial}\). The start and end locations for the navigation of MR while performing tasks is \(r_{1}\). Firstly, MR gathers information of \(r_{2}\) location and return to \(r_{1}\) location after gathering the information of \(m-1\) desired places and the information of \(r_{1}\) location. Information for \(r_{1}\) might be specified as being gathered after it returns in and only gathered once. The MR needs to employ a collision-avoiding approach while traversing across an unstable obstacle. It can decide to halt moving forward, although doing so will lead to the process of obtaining information consuming a longer time. It might additionally accelerate or follow an alternate route, both of which require extra power. Acquisition operations are undefined because of variable obstacles throughout the surveillance domain.