Estimating the Expected Time to Enter and Leave a Common Target Area in Robotic Swarms

Coordination algorithms are required to minimise congestion when every robot in a robotic swarm has a common target area to visit. Some of these algorithms use artificial potential fields to enable path planning to become distributed and local. An efficiency measure for comparing them is the time to complete a task in relation to the number of individuals in the swarm. To compare distinct solutions as the swarm grows, experiments with different numbers of robots must be simulated to form a plot of the function of the task completion time versus the number of robots or other parameters. Nevertheless, plotting it for many robots through simulation is time-consuming. Additionally, the inference of a global swarm behaviour as the task completion time from the local individual robot motion controller based on potential fields and other dynamical variables is intractable and requires experimental analysis. Based on that, equations are presented and compared with simulation data for estimating the expected task completion time of state-of-the-art algorithms, robots using only attractive and repulsive force fields and mixed teams for the common target area problem in robotic swarms with not only the number of robots as input but also environment- and algorithm-related global variables, such as the size of the common target area and the working area, average speed and average distance between the robots. This paper is a fundamental first step to start a discussion on how better approximations can be achieved and which mathematical theories about local-to-global analysis are better suited to this problem.