# Coupled-Error-Based Formation Control for Rapid Formation Completion by Omni-Directional Robots
## 1 Introduction  
The cooperation of multiple robots has been an interesting research topic in the robotics field, as it expands the scope of applications in which robots can be employed and enhances the performance of the target system. Recently, the application of multi-robot systems has increased across various fields, demonstrating commercial capabilities, and prompting more practical products and services with multi-robot systems. This trend has accelerated with advancements in control, networking, and AI technologies. Examples include multi-drone entertainment such as drone shows [1, 2], disinfection and pest control systems using multiple robots, goods movement within logistics centers [3], and military applications such as Manned-Unmanned Teaming (MUM-T) [4, 5]. These innovations represent significant progress in the utilization of multi-robot systems across diverse fields. Successful multi-robot collaboration fundamentally requires formation control, a method that involves controlling multiple robots to form and maintain a specified formation. Since the collaboration of multiple robots typically occurs after the formation has been established, the rapid formation and maintenance of this structure are critical to the success of the mission. As illustrated in commercial applications such as drone shows or the collaborative movement of objects, as well as docking among unmanned vehicles during transit (e.g., AUV (Autonomous Underwater Vehicles)/USV (Unmanned Surface Vehicles) deployment/retrieval and UAV (Unmanned Aerial Vehicles) aerial refueling), maintaining formation between entities is directly linked to the success or failure of the mission. Various multi-agent system-based commercial applications have been supported by the following research outcomes. Extensive studies have been conducted on the leader-follower strategy for building and keeping multiple robot formations. This method involves controlling followers that move in a formation dictated by a leader, as depicted in Figure 1, when the desired position of the followers is generated from the preceding robot. Alternatively, when a follower acts as a leader for the next, it can be called an interconnected system [6]. In this interconnected system, the configuration of the information flow may lead to temporary disruptions in maintaining the formation. If a control algorithm utilizing each follower's tracking control law is applied, the ability to maintain the formation depends on the control performance of individual followers. This strategy can derive the'slinky-type effect', where errors generated during the movement of a preceding robot are transferred and accumulate among subsequent followers [6; 7; 8; 9]. To prevent such phenomena, a leader formation strategy has been proposed [10], where, as depicted in Figure 2a, followers moving based on a preceding agent are provided with the leader's information to avoid the transmission and accumulation of errors. However, even if the entire multi-robot system is maintained, for this control strategy to be applied in real systems, a network system capable of consistently transmitting the leader's information to all followers should be guaranteed. In contrast to the leader-follower technique, a virtual-structure approach has been conducted. This method involves setting a virtual leader [11]. Each follower moves the formation based on the virtual leader. Unlike the leader-follower strategy, the virtual-structure strategy generates the formation of information from a virtual leader, thus preventing the transmission and accumulation of errors. However, this strategy faces the challenge of requiring all followers to share synchronized virtual leader information. With the advent of  
Figure 1: The inter-connected leader–follower formation. Figure 2: Methods preventing slinky-type effect. (**a**) Information flow of leader formation strategy; (**b**) Information flow of bi-directional data acquisition strategy. ultra-low latency and broadband communications such as 5G and Wi-Fi 6, this drawback is being mitigated.