  This paper focuses on spatial time-optimal motion planning, a generalization
of the exact time-optimal path following problem that allows the system to plan
within a predefined space. In contrast to state-of-the-art methods, we drop the
assumption that a collision-free geometric reference is given. Instead, we
present a two-stage motion planning method that solely relies on a goal
location and a geometric representation of the environment to compute a
time-optimal trajectory that is compliant with system dynamics and constraints.
To do so, the proposed scheme first computes an obstacle-free Pythagorean
Hodograph parametric spline, and second solves a spatially reformulated
minimum-time optimization problem. The spline obtained in the first stage is
not a geometric reference, but an extension of the environment representation,
and thus, time-optimality of the solution is guaranteed. The efficacy of the
proposed approach is benchmarked by a known planar example and validated in a
more complex spatial system, illustrating its versatility and applicability.
