# Consensus Algorithms Based Multi-Robot Formation Control under Noise and Time Delay Conditions
## 1 Introduction  
In recent years, with the continuous development of computer science, complex network theory and control theory, autonomous mobile robots have received more and more attention [1]. Compared to single mobile robots, multi-mobile robot systems have better stability, higher fault tolerance and higher work efficiency. As a result, they have better application prospects and higher research value in the fields of reconnaissance, patrol, rescue and environmental survey. Formation control of multi-mobile robots is the basis of multi-mobile robot systems, and has become a hotspot in the field of robotics [2]. As part of the design process of multi-robot formation control algorithm, many problems need to be considered, including robot model, external environmental interference, sensor measurement noise, algorithm control precision, and the controllability of different formations [3]. The existing formation control algorithms for multi-robots mainly include the leader-follower algorithm [4], the behavior-based algorithm [5], the graph theory-based method [6], the virtual structure method [7],and the artificial potential field method. The leader-follower algorithm has flexible motion strategy and scalability, but the algorithm cannot form stable and reliable feedback between the follower and the leader. Therefore, the control error of the follower will increase with interference from the environment. In particular, when the leader fails, it can cause the entire multi-robot system to crash. The behavior-based algorithm can effectively reduce the complexity of the entire formation control algorithm, but it has higher requirements in terms of sensor sensing ability and communication ability between robots, and cannot accurately quantify the behavior of robots during operation. Thus, it is difficult to guarantee the system's robustness using the behavior-based algorithm. The virtual structure method is convenient for designing the formation behavior of multi-robot systems, while due to the constraints of rigid structures, it lacks flexibility with respect to obstacle avoidance and formation transformation. The artificial potential field algorithm has a simple structure and can effectively avoid collisions and obstacles, but it is susceptible to interference when maintaining the formation, and it is difficult to perform precise formation control. Moreover, the potential energy function needs to be reset if the formation transformation is performed, leading to a lack of flexibility. In view of the shortcomings of the traditional formation control algorithm, considering the increase in the number of robots in the multi-robot system and the continuous improvement of the data processing capability of a single robot, the distributed multi-robot control algorithm has attracted the attention of researchers. The distributed multi-robot system can make full use of the data processing resources of the robot and share the pressure of the central processing machine, which has great advantages in terms of flexibility and fault tolerance [8; 9]. In addition, solving the consensus problem is the core of the distributed multi-robot control algorithm [10]. There are already mature and sophisticated strategies for solving the consensus problem in ideal environments [11; 12]. However, in practical applications, uncertain factors like communication noise, communication delay and measurement error will still lead to many problems in multi-robot formation control. Some algorithms have considered some practical problems. Reference [13] studied the conditions of the system reaching consensus under uniform delay, when the communication structures of second-order multi-robot systems were a directed graph with spanning tree or a strongly connected graph, respectively. However, that paper does not consider the noise condition or consensus under different delay conditions. Reference [14] studied the consensus problem of second-order multi-robot systems under noisy conditions.