  UAV trajectory planning is often done in a two-step approach, where a
low-dimensional path is refined to a dynamic trajectory. The resulting
trajectories are only locally optimal, however. On the other hand, direct
planning in higher-dimensional state spaces generates globally optimal
solutions but is time-consuming and thus infeasible for time-constrained
applications. To address this issue, we propose $\delta$-Spaces, a pruned
high-dimensional state space representation for trajectory refinement. It does
not only contain the area around a single lower-dimensional path but consists
of the union of multiple near-optimal paths. Thus, it is less prone to local
minima. Furthermore, we propose an anytime algorithm using $\delta$-Spaces of
increasing sizes. We compare our method against state-of-the-art search-based
trajectory planning methods and evaluate it in 2D and 3D environments to
generate second-order and third-order UAV trajectories.
