# Coupled-Error-Based Formation Control for Rapid Formation Completion by Omni-Directional Robots
## 1 Introduction
Yet, in such cases, a centralized control method using a supervisor that collects and controls information from all agents might be more efficient. These results emphasize the importance of a network configuration capable of sharing information among all agents to address the issues arising during formation control. With recent advancements in communication technology, the practical implementation of such proposals is becoming feasible in manufacturing sites, indoor services, and urban areas. Nonetheless, extensive and broadband network deployment remains unfeasible in many environments such as underwater settings, disaster zones, military operations, and remote or extreme locations like polar regions or space. Therefore, a method that rapidly generates and maintains swarm formations using only the information available from surrounding agents is essential. To overcome the slinky-type effect that occurs when controlling followers' movement along with surrounding agents, a platoon control method based on coupled sliding mode utilizes the information of nearby followers to form and maintain the entire system. By coupling the information of leading and following agents in the controller, the entire system can be rapidly formed, mitigating the slinky-type effect and maintaining the structural integrity of the system. In this scenario, although the overall trajectory convergence of the system may slow down, the advantage lies in the rapid formation of the entire system. However, this approach, which primarily targets a one-dimensional platoon system, has limitations when applied to mobile robots moving in a two-dimensional plane, and the coupling ratio fixed as a constant renders the system rigid. Therefore, this paper proposes a formation control algorithm based on error-coupling that allows for rapid formation and maintenance of a multi-robot system with minimal communication constraints. Furthermore, moving beyond the static coupling ratio proposed in previous works, this paper adopts a flexible coupling ratio that enhances the speed of swarm formation and the trajectory tracking performance of the entire system based on the completeness of the formation. To ensure the stability of the entire system, this paper shows that the coupled error (CE), is bounded and converges to zero, which is the error coupled between the position error from the desired position generated by its leader and the following robot's position error. Additionally, the coupling ratio between the forward and backward errors is represented using the distance between the desired position and the agent. This approach ensures that as the position error of the following agent increases, the formation aspect is more significantly reflected over the trajectory tracking performance, and as the error decreases, the focus shifts back to enhancing the trajectory tracking performance. The contributions of this paper are as follows. Firstly, by coupling the information of surrounding followers, the proposed algorithm enables rapid formation and maintenance of formations, surpassing traditional methods that control each follower independently. Unlike previous approaches where the leader merely follows the trajectory and the followers track the leader--making the completion time of the entire formation dependent on the performance of the followers--the algorithm presented here allows the leader to consider the positions of the followers and move towards them to quickly establish the formation before proceeding along the trajectory. This reduces the formation time, which is crucial for applications requiring swift collaboration start, such as aerial refueling of UAVs and deployment/retrieval of USVs/AUVs, where a slight delay in trajectory following is acceptable for faster mission initiation. The paper is organized as follows: Section 2 describes the system under study, followed by an explanation of the coupled-error-based swarm control in Section 3.