  Time-critical tasks such as drone racing typically cover large operation
areas. However, it is difficult and computationally intensive for current
time-optimal motion planners to accommodate long flight distances since a large
yet unknown number of knot points is required to represent the trajectory. We
present a polynomial-based automatic optimal synthesis (AOS) approach that can
address this challenge. Our method not only achieves superior time optimality
but also maintains a consistently low computational cost across different
ranges while considering the full quadrotor dynamics. First, we analyze the
properties of time-optimal quadrotor maneuvers to determine the minimal number
of polynomial pieces required to capture the dominant structure of time-optimal
trajectories. This enables us to represent substantially long minimum-time
trajectories with a minimal set of variables. Then, a robust optimization
scheme is developed to handle arbitrary start and end conditions as well as
intermediate waypoints. Extensive comparisons show that our approach is faster
than the state-of-the-art approach by orders of magnitude with comparable time
optimality. Real-world experiments further validate the quality of the
resulting trajectories, demonstrating aggressive time-optimal maneuvers with a
peak velocity of 8.86 m/s.
