# Effect of Formation Size on Flocking Formation Performance for the Goal Reach Problem
## 1 Introduction  
Many researchers have focused on robotic swarms--robot groups that co-operate to achieve a global goal in a complex environment [1]. The flocking problem, inspired by the behavior of animals (e.g., birds flocking or fish schooling), is a subfield of swarm robotics [2]. A swarm consists of a set of similar members having the same characteristics and progressing in an asynchronous manner. There is no leader in the swarm directing other members to perform the planned tasks. Thus, a swarm operates in a decentralized context, where complex behavior arises through the labor of autonomous agents that are acting on local information. An agent cannot conclude the swarm aims without collaborating with the rest of the group [3]. One of the main components of a swarming system is the exhibition of cooperative behavior. A flock involves multiple agents that possess real-time group behavior capabilities that facilitate achieving a common target, such as foraging or migration [4, 5]. The design of swarm robotics systems is guided by swarm intelligence principles. [PERSON] [6] divided the swarm behaviors into four major categories--spatial organization, navigation, decision making, and miscellaneous--each of which has different areas of application. [PERSON] [7] proposed an extension to the categorization of the basic swarm behaviors. They used this taxonomy to classify a number of existing swarm robotic applications from research and industrial domains. Several studies were orientated towards modeling swarm robots to increase the understanding of natural swarm systems [8]. In a robotic swarm, the collective behavior results from local interactions among the robots and between the robots and the environment. One area that researchers have investigated isthe study of the efficiency in self-regulation of swarms of birds, schools of fish, and social insect colonies, whereas others initiated models to generate and optimize artificial swarm systems as engineered in the field of swarm robotics [9]. One can notice the importance of models that suggest novel empirical experiments, as well as the prominence of empiric experimentation that results in relevant model parameterizations [10]. [PERSON] et al. [8] published an extensive review presenting a discourse on a large panoply of features, covering swarm robotics. The individual elements in such flocks move, while maintaining specific velocities and distances among each other without splitting. [PERSON] [11; 12] has specified three fundamental flocking rules: collision avoidance, flock centering, and velocity matching (alignment). Animal groups change their structure depending on internal or external stimuli, thus maximizing the fitness of individuals as circumstances change [13]. The behavior of these agents, integrated within the structural order, results in changes of direction and shape without affecting the group's coherence [13]. Formation control aims to produce control commands that are sufficient to drive multiple agents to reach their status constraints [4]. To endow self-organization systems with flexibility and robustness, controllers use decentralized mechanisms [14]. A controller is an essential structure that is used in the implementation and simulation of swarm systems [15]. There are three main approaches to formation control systems [4; 12; 16]: the leader-follower approach [12; 17; 18; 19], the virtual structure approach [1; 20; 21], and the behavior-based approach [5; 14; 16; 22; 23; 24; 25; 26]. Despite the simplicity of the leader-follower approach, it may suffer from being highly dependent on the leader agent and from the large amount of information exchange required between the leader and each of the followers due to being a centralized system [4]. Moreover, virtual structure approaches lack formation modification when the formation changes due to structural redesign, which increases the required computation and degrades collision avoidance abilities.