# Coupled-Error-Based Formation Control for Rapid Formation Completion by Omni-Directional Robots
## 6 Conclusions  
This paper focuses on rapidly configuring formations to support the swift execution of tasks by multiple robots. Accordingly, a CE-based formation control algorithm that utilizes the information of both the leader and followers is proposed for a multi-robot system comprised of omni-directional robots, aiming to achieve rapid formation completion and effective trajectory tracking. Unlike other approaches that rely solely on the desired position information for formation and maintenance, which predominantly depends on the follower's tracking performance, the algorithm proposed in this paper enables the leader to adjust its actions based on the state of its follower, thereby maintaining the overall cluster  
Figure 8: Performance in the clustering control algorithm when applying q based on distance: (**a**) the routes of the member robots; (**b**) the distance between formation and follower; (**c**) coupled errors, \(E_{i}^{x}\) and \(E_{i}^{y}\); (**d**) position error, \(e_{i}^{x}\); (**e**) position error, \(e_{i}^{y}\); (**f**) Changes in q over time.  
As demonstrated in simulation results, by applying the proposed coupled-error-based algorithm to the multi-robot system, the leader not only follows the trajectory but also moves towards the followers upon checking their states, thus quickly completing a formation. Furthermore, unlike conventional algorithms that require high-performance communication to use the entire system's leader information or to simulate a virtual leader, the proposed method swiftly forms clusters by using only the information of nearby entities. Additionally, velocity constraints were implemented in the simulations to validate the proposed algorithm under realistic environmental conditions.  
In this paper, to verify the versatility of the proposed algorithm, the error coupling ratio q was varied, and the performance of the algorithm was evaluated across various aspects by allowing q to be determined based on the degree of cluster formation completion. Moving forward, the coupled-error-based cluster control algorithm introduced in this paper will be applied to actual omni-directional drive systems in practical use cases, and it is also expected to be implemented in other robotic systems with different locomotion capabilities.  
Conceptualization, J.-W.K.; Methodology, J.-W.K. and H.K.; Software and Simulation, H.K. and S.-J.; Validation, J.-W.K. and H.K.; Formal analysis, J.-W.K.; S.-J.Y. and Y.-H.C.; Investigation, G.C. and Y.-H.C.; Data curation, J.L. and N.-H.L.; Writing--original draft, J.-W.K.; Writing--review and editing, J.-W.K.; Supervision G.C. and Y.-H.C.; Formal analysis, G.C. and Y.-H.C. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the Korean Planning & Evaluation Institute of Industrial Technology (KEIT) and conducted by the Ministry of Trade, Industry and Energy (MOTIE) (Robot Industrial Core Technology Development Project, Project Number 20018764).  
Not applicable.  
The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.  
The authors declare no conflicts of interest.