## 1 Introduction
Reference [13] proposed a formation control method that combines ESO with distributed model predictive control (DMPC). In response to the problem of a multiple AUV formation in environments with interference, [PERSON] also adopted a method that combined the ESO with the DMPC [4]. However, unlike reference [13], [PERSON] et al. directly took the state of the robot observed by the ESO as the input of the DMPC and proposed a Lyapunov-based model predictive control to ensure the feasibility and stability of the algorithm. According to above studies, regarding the tracking control, plenty of controllers can be designed using the backstepping control, the sliding model control (SMC), the adaptive neural network control, and the model predictive control (MPC). The backstepping algorithm adopts a reverse reasoning approach. In a hypothetical virtual system, the algorithm links the changes in the robot state at each sampling time and the adaptive adjustment function of uncertain parameters to the designed Lyapunov function, as well as gradually corrects the deviation between the current state and the expected state, thus achieving tracking control. The sliding mode control is also a very popular algorithm for trajectory tracking due to it not only possessing the trait of model independence, but that it also has the trait of good anti-interference performance. However, the control quantity obtained by the controller has a high-frequency chattering phenomenon, which is not conducive to the execution of actuators. The neural network algorithm is another commonly utilized algorithm in underwater tracking control [14; 15]. The neutral network-based tracking control is insensitive to the dynamic model of the robot, and it owns strong adaptability and learning capability. However, due to changeable underwater environments, the training and learning process are difficult, which leads to a poor real-time performance in the tracking algorithm [16; 17]. Trajectory tracking controllers based on a backstepping controller, sliding mode controller, and an adaptive neutral network control algorithm could not achieve optimal control performance with various constraints that include the inherent constraint of robot itself, such as the saturation and dead zone of thrusters, as well as some other constraints that are caused by the actual tasks, such as the state value threshold of the robot. In contrast, MPC is known as one of the few control methods that can explicitly handle system constraints systematically and can optimize control performance [18]. Usually, the MPC contains linear MPC and nonlinear MPC. The nonlinear MPC has more accurate tracking effects, as well as a higher cost of computation than linear MPC. A comparison of the above-discussed control methods is shown in Table 1.  
This article aims at proposing a formation tracking control that could be carried out and deployed easily on small spherical robots. The small spherical robot is born with some complex problems such as limited energy, computational ability, and perception ability; an inaccurate dynamic model; various constraints; etc. In order to deploy the formation algorithm to small spherical robots, the above issues must be considered. In addition, another challenge of formation control for small spherical robot lies in the avoidance of inner collisions. As such, some methods must be designed to overcome the above problems. The main contributions of this article are as follows:  
1. The multiple small spherical robot formation problem is split into trajectory tracking problems. The virtual structure is designed for obtaining the reference trajectory of each the robots in a formation. A model predictive-based formation tracking controller is proposed, and the ESO is designed to deal with the inaccurate dynamic model and other disturbances. 2. To achieve the inner collision avoidance, a collision cost term is designed.