## 1 Introduction  
Charging sensor nodes (nodes) with a wireless mobile vehicle (vehicle) is a crucial aspect of wireless rechargeable sensor networks (WRSNs) since it can enhance multiple aspects of network performance. Given the limited residual energy of nodes, it is essential to develop charging scheduling for vehicles as a top priority in WRSNs. Nodes typically transmit data to the base station via one or multiple relay nodes, as specified by the routing protocol [1]. As a result, nodes that are geographically closer to the base station often have to forward significantly more data from remote nodes, which consumes additional energy. Failure to charge these nodes on time can lead to the energy hole phenomenon [2, 3],negatively impacting the network's normal operation. Significant research efforts have been devoted to addressing this issue, including fixed vehicle trajectories [4; 5; 6; 7; 8] in periodical schemes. In these schemes, the trajectory is predefined; the vehicle travels along the trajectory to charge each node in a network. However, due to a lack of flexibility, this approach is unable to fully address the problem. A promising approach to maintaining the perpetual operation of WRSNs is to plan dynamic trajectories for vehicles using an on-demand architecture [9; 10; 11; 12; 13; 14; 15; 16] that provides control over the trajectory. In general, nodes send charging request packets to the base station when their residual energy falls below a preset threshold. Vehicles then perform the charging tasks along a dynamic trajectory determined by the base station according to a scheduling scheme. This dynamic trajectory planning is considered as charging scheduling, which not only reduces energy consumption during vehicle movement, but also extends the network's lifetime to some degree. However, charging nodes solely with vehicles has some disadvantages, including low speed and offroad limitations. These disadvantages can negatively impact the continuous operation of the network when it is deployed in a large and complex terrain region under heavy load, as numerous simultaneous charging requests cannot be met. Recently, the rapidly growing application of drones is advancing to a new paradigm, which considers charging nodes with wireless charging drones (drones) carrying a wireless charging device [17; 18; 19; 20; 21; 22]. Drones have high flight speed, more than five times that of vehicles [23]. Additionally, they can fly over obstacles without having to take a detour. Although drones can overcome the disadvantages of vehicles, their small battery capacity still limits the flight distance and the number of charging nodes, thereby restricting their application in a large sensing region. After comprehensively considering the merits and demerits of vehicles and drones, we first propose a new WRSN model based on on-demand architecture that includes one vehicle, multiple drones, one base station and a set of nodes within a specified sensing region. In this scheme, it is assumed that the base station will plan the most efficient hybrid charging scheduling for both the vehicle and the drones at the same time. However, the novel WRSN model still faces some new challenges. First, an optimal scheduling for one vehicle for visiting predetermined places usually is an NP-hard problem [13; 24]. Moreover, scheduling one vehicle and multiple drones at the same time is even more complicated and challenging, and it is hard to find an optimal solution in an efficient way. This work presents a charging strategy named the long distance first matching (LDFM) algorithm to collaboratively schedule a vehicle and multiple drones in an efficient way. First, LDFM matches nodes with charging requests according to the distance between nodes and their deadline, then allocates the nodes that require the long distance to drones by applying a matching algorithm. The matching algorithm chooses the maximum matching greedily according to the number of drones carried by the vehicle.