Optimal Virtual Tube Planning and Control for Swarm Robotics

This paper presents a novel method for efficiently solving trajectory planning problems for swarm robotics in cluttered environments. While recent research has demonstrated high success rates in real-time local trajectory planning for swarm robotics in cluttered environments, optimizing every trajectory for each robot is computationally expensive, with a computational complexity of $O\left(n^2\right)$ to $ O\left(n^3\right)$. To address this issue, we first propose the concept of the \emph{optimal virtual tube}, which includes infinite optimal trajectories. Under certain conditions, any optimal trajectory in the optimal virtual tube can be expressed as a convex combination of a finite number of optimal trajectories, with a computational complexity of $O\left(1\right)$. Afterward, a planning method of \emph{the optimal virtual tube} is proposed. In simulations and experiments, we show that the proposed method efficiently reduces calculation and is validated by comparison with traditional methods.