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.

翻译：本文提出了一种在杂乱环境中高效解决集群机器人轨迹规划问题的新方法。尽管近期研究已展现了在杂乱环境中实现集群机器人实时局部轨迹规划的高成功率，但为每个机器人优化每条轨迹的计算成本高昂，其计算复杂度为 $O\left(n^2\right)$ 至 $O\left(n^3\right)$。为解决这一问题，我们首先提出了“最优虚拟管道”的概念，该管道包含无限条最优轨迹。在特定条件下，最优虚拟管道中的任意最优轨迹均可表示为有限条最优轨迹的凸组合，且计算复杂度为 $O\left(1\right)$。随后，我们提出了一种“最优虚拟管道”的规划方法。仿真与实验表明，与传统方法相比，所提方法有效降低了计算量，并验证了其有效性。