## 5 Conclusions  
In this paper, we have proposed a circular formation controller with obstacles avoidance for a group of multiple nonholonomic mobile robots to collaboratively encircle and track a static or moving target in an environment with obstacles. The controller includes three sub-parts, where the first is circular motion control law by using the idea of circular trajectory tracking control that each robot moves to the desired circular around a target, maintaining a preset distance between the robots and the target; the second is phase positioning and spacing assignment control law by introducing a nonlinear function, for which each robot can determine the initial phase on the circle itself and maintain the desired relative angular spacing from its neighbors; and the third is obstacle-avoidance control law, which only considers the repulsive potential field, for which each robot is able to effectively avoid obstacles in the complicated environment while maintaining the stability of the formation. Then, theoretical analysis has been provided to show the convergence of  
Figure 9: Experimental results of case 3. **(a)** The motion views of three nonholonomic mobile robots forming a evenly distributed circular formation with same circular radius to enclose and track a moving target with time-varying speed. **(b)** Trajectories of the three robots and the target in the plane. **(c)** Error between current phase and the desired phase of robot 1. **(d)** Distance errors between the three robots and their desired circles centered on the target. **(e)** Errors between current angular spacing and the desired angular spacing of all adjacent inter-robots. **(f)** The linear speeds of the three robots. **(g)** The angular speeds of the three robots.  
the system under the proposed controller. Finally, the simulation and experimental results demonstrate good control performance of the proposed control scheme. The controller can realize multiple nonholonomic mobile robots with speed constrains fencing around and tracking a static or moving target in an environment with obstacles, which has great practical value for engineering applications.  
In future work, our proposed control scheme should be implemented in a local frame that is more in line with practical engineering so that only local information is utilized without knowing global information. In addition, the position and velocity of the target are difficult to obtain directly in practice, especially in the global frame. It is necessary to design an estimation law so that each robot can estimate the position and velocity of the target.  
Conceptualization, X.L. ([PERSON]) and S.H.; Data curation, X.L. ([PERSON]); Formal analysis, G.W.; Funding acquisition, G.W.; Investigation, S.H.; Methodology, X.L. ([PERSON]) and G.W.; Project administration, G.W.; Resources, X.L. ([PERSON]); Software, X.L. ([PERSON]); Validation, X.L. ([PERSON]), G.W., C.S. and H.C.; Writing--original draft, X.L. ([PERSON]); Writing--review and editing, X.L. ([PERSON]). All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Science and Technology Project of Fine and Rescue Bureau of Emergency Management Department, Grant No. 2020 XFZD15.  
Not applicable.  
We would like to thank Beijing University of Posts and Telecommunications for providing the experimental site and the cooperative unit China Academy of Safety Science and Technology for providing project funds.  
We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled, \"Cooperative Target Enclosing and Tracking Control with Obstacles Avoidance for Multiple Nonholonomic Mobile Robots\".