  With much research has been conducted into trajectory planning for
quadrotors, planning with spatial and temporal optimal trajectories in
real-time is still challenging. In this paper, we propose a framework for
generating large-scale piecewise polynomial trajectories for aggressive
autonomous flights, with highlights on its superior computational efficiency
and simultaneous spatial-temporal optimality. Exploiting the implicitly
decoupled structure of the planning problem, we conduct alternating
minimization between boundary conditions and time durations of trajectory
pieces. In each minimization phase, we leverage the algebraic convenience of
the sub-problem to escape poor local minima and achieve the lowest time
consumption. Theoretical analysis for the global/local convergence rate of our
proposed method is provided. Moreover, based on polynomial theory, an extremely
fast feasibility check method is designed for various kinds of constraints. By
incorporating the method into our alternating structure, a constrained
minimization algorithm is constructed to optimize trajectories on the premise
of feasibility. Benchmark evaluation shows that our algorithm outperforms
state-of-the-art methods regarding efficiency, optimality, and scalability.
Aggressive flight experiments in a limited space with dense obstacles are
presented to demonstrate the performance of the proposed algorithm. We release
our implementation as an open-source ros-package.
