## 1 Introduction
### Related Work  
Most of the studies related to node replacement in WSNs have handled these problems by assuming that they have previous knowledge of the node or event position, or even the node energy consumption [13; 14; 15; 16; 17; 18]. In [1], three policies were proposed. Each policy determined the importance of each failed node on the coverage and lifetime of the network by assigning a weight to each failed node. If the weight was greater than the policy threshold, the node was replaced; otherwise, it was ignored. These policies aimed to maximize the lifetime and simultaneously minimize redundant resources. The weight of the network knowledge should be evaluated, and a central server was required to calculate them. In [8], knowledge of the events was needed, and the position of redundant nodes was known because they were stored in the sink. The proposed scheme scheduled the travels of the robot and duty cycles of the sensors. The goal was to maintain coverage while minimizing the traveled distance. However, the scheme only considered point coverage, similar to that proposed by [13] (in which area coverage is considered). The nodes were grouped in sets, a staircase-based scheduling model was implemented, and a fixed number of backup nodes were deployed in sets. Authors in [19; 20] have studied the network to provide adequate quality of service (using energy replenishment and data collection) through limited mobile devices in WRSNs with multiple sinks. Scheduling the mobile devices' recharging and data collection capabilities ensures that the total weight of the recharged sensors can be operated without time limits being maximized. The mobile robot is utilized to perform replacement and recharging tasks. To the best of our knowledge, researches have not explored the benefit of the airplane robot (Air-R) replacement from the R-WSN perspective (utilizing Air-R replacement). Nevertheless, significant research in the sensor network community has explored high replacement efficiency using mobile robots [21]. Some of the mobile robot problems are the large travel distance, the exhaustive search process to find the failure node which must be replaced, and the harsh environment in which it works. Therefore, the proposed method in this paper minimizes the travel distance and pressure work of the mobile robot effectively, deals with any failure of the network, and solves problems before expansion. Sensor nodes are deployed regularly and at random in a circular area, which is divided into multiple rings with a constant width [22]. A mobile robot is utilized to recharge WSNs at the circular trajectory of the mobile robot (to the energy consumption rate of each sub-region of the network). This study investigates how the mobile charger (MC) traverses and transfers energy to the sensor node in a wireless method. This strategy can significantly reduce the total death ratio of nodes compared with traditional WSNs. Some researchers have worked on the distribution of sensor nodes to prolong the lifetime of sensor networks [23], presenting a novel edge-based routing protocol called BeamStar for WSNs. The authors divided the network into rings and sectors. To provide location information for the nodes, the power-controlled capability of the base station (BS) was utilized to scan the network with different power transmission levels. In [24], another routing protocol for edge-based WSNs was presented called the cluster-based BeamStar (CBS). The CBS was introduced to overcome some of the disadvantages of BeamStar. The CBS uses the same technique as BeamStar but with a developed scan process to provide sensor nodes with location information. This protocol is successful in using network resources optimally and in reducing internode communication load. The CBS is explained in three different phases, namely, locating, cluster-building, and data-transmitting phases. In the first phase, the entire field is scanned using different transmission power levels (R).