  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.
