# Coupled-Error-Based Formation Control for Rapid Formation Completion by Omni-Directional Robots
## 4 Simulation Results  
For the numerical simulations, we consider the multi-robot system in Figure 4, each of which consists of two robots--a leader and follower. Here, the 0 th robot represents the entire system's trajectory, the 1 st robot is the leader, and the 2 nd robot is the follower. The formation is built using \(L=1\) and \(\psi=\pi/2\). The initial positions of each robot, \((x_{i}(0),y_{i}(0),\theta_{i}(0))\) (m, m, rad) (\(i=0,1,2\)), are (0, 0, 0), (\(-4\), 2, 0), and (\(-8\), 2, 0), respectively. The trajectory moves following the velocity profile. \[v_{0}^{x}=0.3\ (\mathrm{m/s}),\ v_{0}^{y}=0.0\ (\mathrm{m/s}),\ 0 \leq t<20\ (\mathrm{s})\] \[v_{0}^{x}=0.0\ (\mathrm{m/s}),\ v_{0}^{y}=0.3\ (\mathrm{m/s}),\ 20 \leq t<40\ (\mathrm{s})\] \[v_{0}^{x}=-0.3\ (\mathrm{m/s}),\ v_{0}^{y}=0.0\ (\mathrm{m/s}),\ 40 \leq t\ (\mathrm{s})\]  
Additionally, considering the real system, the analysis applies velocity saturation constraints of \(-1<v_{i}^{x}<1\) and \(-1<v_{i}^{y}<1\). In this paper, a system that only uses the information of the preceding robot without coupling ratio \(q\) is configured to serve as the control group for comparing and analyzing the performance of the proposed clustering control algorithm. The control algorithm of the system used as the control group representing previous research has the form of a linear controller, as follows. \[v_{i}^{x}=v_{i-1}^{x}+k_{x}\epsilon_{i}^{x},\ v_{i}^{y}=v_{i-1}^{y}+k_{y} \epsilon_{i}^{y} \tag{19}\]  
where the control gain, \(k_{x}>0\) and \(k_{y}>0\) are constant values. The linear controller in Equation (19) is used to compare the performance of the controller that uses only the information of the preceding robot with that of the proposed controller. First, after applying \(q=5\), the tracking performance of the proposed formation control method in Scenario 1 is presented in Figures 5 and 6. Figure 5 shows the routes of the member robots. Figure 5a shows the route of a multi-robot system using a controller that only follows its own desired position. Since it does not consider other robots, it can be observed that the formation is created after the follower moves a long distance. Compared to the single robot-based formation control algorithm in Figure 5a, Figure 5b shows that the leader, with the proposed CE-based formation control algorithm applied, moves towards the follower to form a cluster formation and then follows the trajectory. As can be seen in Figure 5a, the formation is quickly established after the follower travels a short distance. Figure 6 shows the detailed performance of error convergence and boundedness. From Figure 6a-c, we can see the coupled error and position errors of each robot are bounded and converge to zero. In Figure 6c, it is evident that the leader moves towards the follower between \(t=0\) and about \(t=8\) s, even at the expense of an increased trajectory tracking error. This highlights that, despite the increase in error, the leader prioritizes moving towards the follower. Figure 6d shows the distance between the follower robot and its desired position, which means the completeness of the formation. As can be seen in Figure 6d, it can be ensured that the distances between robots within the formation converge more quickly in the system with the proposed CE-based formation control algorithm applied. To analyze the effect of the coupling ratio \(q\), the performance of the entire system is observed by varying \(q\) (q = 1, 5, 10, and 50). Figure 7 shows the changes in system performance as \(q\) varies. In Figure 7, we can see that as the coupling ratio \(q\) increases, the leader moves closer to the follower. Conversely, as \(q\) increases, the performance of trajectory tracking decreases. Especially, Figure 7d shows the distance between the desired position and the follower. We can see that increasing q results in faster completion of the formation. As observed in Figures 5-7, the speed of completion of formation and the leader's trajectory tracking performance are determined by the coupling ratio, \(q\).