Recently, the concept of optimal virtual tube has emerged as a novel solution to the challenging task of navigating obstacle-dense environments for swarm robotics, offering a wide ranging of applications. However, it lacks an efficient homotopic path planning method in obstacle-dense environments. This paper introduces Tube-RRT*, an innovative homotopic path planning method that builds upon and improves the Rapidly-exploring Random Tree (RRT) algorithm. Tube-RRT* is specifically designed to generate homotopic paths for the trajectories in the virtual tube, strategically considering opening volume and tube length to mitigate swarm congestion and ensure agile navigation. Through comprehensive comparative simulations conducted within complex, large-scale obstacle environments, we demonstrate the effectiveness of Tube-RRT*.

This paper presents a novel method for efficiently solving a trajectory planning problem for swarm robotics in cluttered environments. Recent research has demonstrated high success rates in real-time local trajectory planning for swarm robotics in cluttered environments, but optimizing trajectories for each robot is still computationally expensive, with a computational complexity from $O\left(k\left(n_t,\varepsilon \right)n_t^2\right)$ to $ O\left(k\left(n_t,\varepsilon \right)n_t^3\right)$ where $n_t$ is the number of parameters in the parameterized trajectory, $\varepsilon$ is precision and $k\left(n_t,\varepsilon \right)$ is the number of iterations with respect to $n_t$ and $\varepsilon$. Furthermore, the swarm is difficult to move as a group. To address this issue, we define and then construct the 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 $O\left(n_t\right)$. Afterward, a hierarchical approach including a planning method of the optimal virtual tube with minimizing energy and distributed model predictive control is proposed. In simulations and experiments, the proposed approach is validated and its effectiveness over other methods is demonstrated through comparison.