## 6 Conclusions  
In this paper, a virtual-structure-based formation control design is proposed for a group of multiple mobile robots with the consideration of obstacle avoidance, where \(N\) basic desired trajectories are generated in advance and parameterized by the path parameter \(s\) to maintain the formation for multiple mobile robots. To achieve collision avoidance, a trajectory generator with a potential function is designed that can adjust the basic trajectories and replan the reference trajectories. Subsequently, a tracking controller is designed to achieve the good tracking performance of mobile robots, and the asymptotical stability is guaranteed via the Lyapunov theorem. A comparative simulation with three sets is implemented with the design of an objective function \(F_{obj}\) to evaluate the tracking performance, and the simulation results verify the good formation and tracking performance of the proposed formation control design for multiple mobile robots with obstacle avoidance and smaller tracking errors. Finally, the real robot experiment is implemented to further verify the effectiveness of the proposed control design.  
Since the potential function-based path planning approach has essential drawbacks, such as the robot group possibly becoming stuck in a deadlock, the path parameter P2 proposed in this paper is developed to further improve the deadlock situation. In detail, the controller C1-P2 can adjust the path parameter to lower the weight of formation at this time, which maintains the good tracking and formation of the rest of this group with the sacrifice of stopping the motions of the stuck robot. However, this deadlock situation cannot be thoroughly solved because the stuck robot is left behind. Meanwhile, due to the hardware condition, the obstacle set is static, which can be relatively easily detected via the OptiTrack cameras. However, in practice, since a collision is usually instantaneous and time-varying, the online perception of obstacles and the identification of obstacle shapes becomes quite challenging, which may lead to a performance reduction in a real environment. Therefore, in future, other collision-avoidance algorithms and the online perception of obstacles and their shapes will be investigated to further improve the tracking and formation performance in the practical environment.  
Conceptualization, X.C. and Z.C.; methodology, X.C. and F.H.; software, X.C. and Y.Z.; validation, F.H., Y.Z. and S.L.; formal analysis, F.H.; investigation, X.C.; resources, S.L. and Y.N.; data curation, S.L. and J.T.; writing--original draft preparation, X.C.; writing--review and editing, X.C.; visualization, S.L. and S.Z.; supervision, S.L. and Z.C.; project administration, S.L., J.T. and S.Z.; funding acquisition, Z.C., Y.N. and S.L. All authors have read and agreed to the published version of the manuscript.  
This work was funded by the Natural Science Foundation of Zhejiang Province, China (No. LY19E050016), National Natural Science Foundation of China (No. 51809235 and No. 61603332), and the Strategic Priority Research Program of the Chinese Academy of Sciences (No. XDA22040202).  
The authors declare no conflict of interest.