###### Abstract
Then, the research group proposed a circle formation control method based on the idea of limit-cycle design acting on the double integrator dynamic model [24]. The designed controller is much more universal because all the desired angular spacings between neighboring agents and the desired distances between each agent and the target are equal nor the requirement. However, only the problem of collision avoidance between agents was considered, and the fact that there may be other obstacles in the actual environment was ignored. For the latter, cooperative following generally addresses some specific missions that do not need to rotate around the goal for energy saving or other practical purposes, such as escorting mission of ships to fleets, and UAVs constantly monitoring specific areas at the specified locations. Similarly, some control strategies have been proposed for cooperatively following a target or an area of interest by a group of agents. In [25], a centralized control algorithm based on the null-space-based behavioral (NSB) control approach has been proposed to deal with the problem of entrapping and escorting an autonomous target by a multi-robot system. The proposed solution was structurally robust to the loss of the vehicles. Similarly, the NSB control method has been defined common tasks in [26], and the Lyapunov theory was used to analyze their stability. For the escorting mission of Euler-Lagrange systems, [PERSON] et al. [27] has structured an outer-inner loop control framework, in which the outer loop considered obstacles, made use of a NSB control architecture and the inner loop applied the adaptive proportional derivative sliding mode control (APD-SMC) law to improve convergence speed and robustness. However, the algorithms required that all robots were evenly distributed on the surface of a sphere or circle and maintained the same specified distance with respect to the target and thus are not suitable for many complex environments and tasks. However, most aforementioned references assume that the target is in a single state of motion instead of all possible motion states; the robots are used to achieve a circular formation on the same circle and evenly distribute on the common circle. In addition, the obstacles avoidance problem of circular formation added, which includes adjacent agents and surrounding obstacles, is rarely considered. Knowing how to integrate the obstacles avoidance algorithm into the circular formation while ensuring the stability of the formation is also a challenge. In this paper, we study the general following control problem in which a team of nonholonomic mobile robots with speed constraints can form desired circular formations with different circle radii and different angle distances to enclose and track a stationary or moving target. The main contributions of the work can be summed up in three aspects:  
1. A novel circular motion control law based on the idea of circular trajectory tracking control is proposed in order to guide multiple nonholonomic mobile robots to converge onto the prescribed circles added with the same or different radii around a static or moving target. On the whole, the designed control law is not only simple and effective, but also can ensure that multiple robots at any initial locations in a plane quickly form the desired circular formation. 2. A phase positioning and spacing adjustment control law has been taken into account by introducing a nonlinear function, maintaining the desired angular spacing from its front neighbors and fixed-phase of the robots on the circles. Hence, by combining it with the circular motion controller, the robots can move to the arbitrary position of the circles given by the user. 3. To solve the obstacles avoidance problem of the multi-robot system in the practical environment, the most well-known artificial potential field method, only with the repulsive force, is adopted to ensure quick obstacles avoidance for each single robot. The remainder of the paper is organized as follows.