## 1 Introduction  
With the rapid development of wireless sensor networks (WSNs), the accurate deployment of a large number of wireless sensors would become an important issue in the near future, e.g., the application of unmanned aerial vehicles (UAVs) or autonomous underwater vehicles (AUVs) [1, 2]. In recent years, many researchers have focused on the problem of position optimization in WSNs since node deployment significantly affects both system lifetime and energy consumption in a two-dimensional (2D) environment. It was proved that a hexagonal structure is the best topology in a 2D network. It provides the maximum coverage area with the minimum number of sensor nodes and minimum energy consumption [3]. The virtual-force algorithm (VFA) is one of the most popular mechanisms for node deployment, in which sensors are moved by the virtual force determined by the relative position of neighbor nodes and the presence of an event [4, 5, 6]. The VFA presents a force-directed approach to improve network coverage with limited sensor nodes. It has low computation complexity with less computation time and one-time repositioning for the sensor nodes. [PERSON] and [PERSON] improved the virtual-force algorithm by adding certain constraints to the virtual force function [7]. Based on the theory of virtual potential field and disc packing, [PERSON] and [PERSON] used attractive and repulsive forces to determine the virtual motion paths and movement rates of the sensor nodes. A probabilistic target-localization algorithm was presented to enhance field coverage [5; 8]. [PERSON] et al. reported a serialized VFA, like the \(L_{max}\)-serialized VFA and the \(D_{th}\_L_{max}\)-serialized VFA. It has good coverage, connectivity, and fault-tolerance to increasing the robustness of a robot sensor network [9]. In order to quickly respond to sudden events or disasters, [PERSON] et al. investigated a distributed relocation method with regular tessellation of the geographical area [6]. [PERSON] et al. introduced a newly algorithm based on Van der Waals force [10], which yielded good uniformity, field coverage, and system convergence. In general, the fixed coverage area and model parameters are provided as the system inputs to the VFA, which cause a flexible and efficient solution in real WSN applications. However, the node location affects the calculations of node velocities and accelerations. In such a case, location accuracy is important in common VFAs. It is necessary to further discuss the effects of system parameters on the localization of sensor nodes. [PERSON] et al. implemented the algorithm computation and analyzed the details of influence of experimental parameters on [PERSON]'s algorithm [11]. They also presented a self-consistent method by leveraging the phenomena in which dusty particles can automatically form hexagonal topology. The influence of the computation scale and shielding length on this algorithm were also investigated [12]. Recently, we presented a virtual-force algorithm inspired by spring force (VFA-SF), in which the sensors are moved by the virtual spring force, determined by the relative position of neighboring nodes. After statistically analyzing the stability of the VFA-SF, we adopted an optimization solution (VFA-SF-OPT) where node deployment begins from the center region by adding an external force to the most peripheral nodes of the network. Simulations showed that the VFA-SF-OPT finally yields perfect hexagonal topology [13]. However, in our previous paper, the theoretical exercises were carried out as proof of concept in the VFA-SF and the VFA-SF-OPT. In the simulated scenarios, the entire area was covered. It is still important to extensively discuss how parameters can potentially affect the performance of node deployment. This paper mainly aims at answering three questions: (1) Since location accuracy is related to node velocity in the algorithm process, what is the influence of node velocity on network topology and system convergence?