# Cooperative Path Planning for Multiple Mobile Robots via HAFSA and an Expansion Logic Strategy
## 1 Introduction  
Mobile robots can be equipped with different sensors and tools to afford a variety of services such as home care, mining detection and object handling [1; 2]. One of the fundamental issues with mobile robots performing tasks is ensuring that they can navigate safely in an unknown indoor environment. Therefore, path planning is crucial for the successful application of mobile robots. The goal of mobile robot path planning is to find a motion path from a starting position to a target position in an environment with obstacles [3; 4; 5]. For the past two decades, there has been a great deal of research on the problem of mobile robot path planning. For example, a novel motion map was constructed for mobile robots based on the BIE (boundary integral equation) method, and then, a point-to point path planning problem was addressed in a known environment with static obstacles [6]. Furthermore, an improved three-dimensional-like grid map was developed to represent the environment model [7], and then, a simple but efficient path planning algorithm was presented to solve robot navigation problems in a static environment. The authors designed an autonomous multi-goal navigation system for picking up or delivering tasks in mobile robotics and a multi-goal path planning method based onthe Lin\(-\)Kernighan heuristics (LKH) algorithm for intelligent service mobile robots in Reference [8]. There are also some intelligent methods that can be applied to mobile robot path planning. The cross probability and the mutation probability for GA (genetic algorithm) were improved and the improved algorithm was applied to the path planning problem of mobile robots in Reference [9], whereas the authors proposed an intelligent motion planning and navigation method for omnidirectional mobile robots via a fuzzy logic algorithm in Reference [10]. Moreover, the developed navigation system is especially suitable for real-time path planning applications. A novel optimal hierarchical global path planning method for mobile robots in a cluttered environment was presented in Reference [11]. In this method, a combination of the triangular decomposition approach, constrained multi-objective PSO (particle swarm optimization) and [PERSON]'s algorithm is presented in order to obtain an optimal path planning trajectory. In addition, due to good feedback information and better distributed computing, the authors proposed a path planning method for mobile robots via an improved ant colony algorithm in grid maps in Reference [12]. Using multiple mobile robots rather than a single mobile robot can improve working capability and performance. Therefore, recently, research on multiple mobile robots has become a hot topic. In previous studies, only static obstacles in the unknown environment were considered. For holonomic wheeled mobile robots in static environments, an optimal multiple mobile robot path planning method based on adaptive charged system search (CSS) algorithms was addressed [13]. However, path planning methods in an unknown environment with dynamic obstacles are even more acute in multiple mobile robot areas. In Reference [14], a new path planning approach for coordinating multiple mobile robots was presented and the authors developed an online strategy to adjust path planning for avoidance of dynamic obstacles. Furthermore, a biologically inspired neural-network-based intelligent method was proposed for a multiple robot system with moving obstacles [15]. The proposed method could plan the paths of multiple robots to avoid collision with dynamic obstacles. Although these previously-developed navigation algorithms have shown good performance for solving robot path planning, they have also shown some limitations such as slow convergence and a local optimum. The local optimal problem is the most common problem in solving path planning. Motivated by the aforementioned reasons, we reconstructed an analytical real-time cooperative navigation algorithm to accommodate multiple mobile robot systems.