# Consensus Algorithms Based Multi-Robot Formation Control under Noise and Time Delay Conditions
## 5 Physical Experiment Verification  
To verify the proposed formation control algorithm, we did the experiment based on a pre-constructed multi-mobile robot research platform built by our laboratory, which was constructed with a self-designed three-wheeled omnidirectional robot carrying an UWB (Ultra-Wide Band) ranging module. The system is shown in Figure 7, the omnidirectional robot is shown in Figure 8, and the performance parameters are shown in Table 1. Because the consensus of the second-order system is analyzed in the theoretical analysis part, the speed and position are consistent, and while the omnidirectional robot is a fully driven robot, the horizontal and vertical directions can be controlled separately. Because the velocity control in a given direction is a second-order system, therefore, multi-omni-directional mobile robots can be decomposed into two one-dimensional second-order multi-robot systems. Therefore, omni-directional robots are used to verify the proposed algorithm. In the experiment, it is possible to determine whether the algorithm is valid based on whether the speed and the position of the robot after final stabilization are consistent. In the data acquisition part, the external positioning data of the robot are collected by the UWB positioning system built by myself, and the speed of the robot itself is collected by the encoder on the wheel of the robot and transmitted to the central processing computer via Wi-Fi for processing. The ranging error between the UWB ranging modules used in the experiment is 7 cm. Experiments were carried out in an indoor environment with length \(\times\) width of 4 m \(\times\) 5 m in order to verify the effectiveness of the proposed algorithm. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter Name** & **Value** \\ \hline weight & 1.5 Kg \\ diameter & 235 mm \\ Maximum linear velocity & 1.2 m/s \\ Maximum angular velocity & 6.6 rad/s \\ Battery capacity & 2800 mah \\ No-load Maximum Standby Time & 2 h \\ Maximum Load Weight & 3 Kg \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Omnidirectional robot parameters. Figure 6: Experiment 2 Group 2 simulation results;\(x\) is position, \(v\) is velocity. (**a**) Trajectory of \(x\) changing with time; (**b**) Trajectory of \(v\) changing with time. It should be pointed out that when the proposed algorithm is applied to a practical multi-robot system, it is necessary to first determine the maximum communication delay between robots and the maximum amplitude of the noise environment, and then design \(k_{1}\) and \(k_{2}\) based on this. At the same time, it should be noted that this experiment mainly focuses on verifying whether the system can achieve consensus under the control law. The collision avoidance behavior of the multi-robot system is not the emphasis in this research. Therefore, the collision avoidance algorithm program is written in the bottom control program of the robot in this experiment. When the robot is about to collide, the formation algorithm program will be interrupted, and the collision avoidance behavior will be executed. When a safe distance between the robots has been reached, the formation algorithm program will continue to be executed [17; 18]. The communication topology used in the experiment is shown in Figure 4. The adjacent communication weight \(a_{ij}\) is \(1\), so \(k_{1}\in(0,10k_{2}^{2})\). The central processor logs on each robot remotely through SSH, and obtains the communication delay between two robots whose communication weight A is not zero by PING command. The time delay between robots in the actual communication environment is time-varying, so take its maximum delay. We get \(\tau_{a1}=0.56s\), \(\tau_{a2}=0.043s\), \(\tau_{a3}=0.047s\), \(\tau_{a4}=0.061s\). The time taken for each robot to receive data and process them is \(\tau_{b1}=0.021s\), \(\tau_{b2}=0.02s\), \(\tau_{b3}=0.021s\), \(\tau_{b4}=0.021s\). Therefore, the time delay between robots is \(\tau_{1}=0.077s\)  
Figure 8: Omnidirectional robots. (**a**) Top view; (**b**) Side view.