  This paper presents a two-stage prediction-based control scheme for embedding
the environment's geometric properties into a collision-free Pythagorean
Hodograph spline, and subsequently finding the optimal path within the
parameterized free space. The ingredients of this approach are twofold: First,
we present a novel spatial path parameterization applicable to any arbitrary
curve without prior assumptions in its adapted frame. Second, we identify the
appropriateness of Pythagorean Hodograph curves for a compact and continuous
definition of the path-parametric functions required by the presented spatial
model. This dual-stage formulation results in a motion planning approach, where
the geometric properties of the environment arise as states of the prediction
model. Thus, the presented method is attractive for motion planning in dense
environments. The efficacy of the approach is evaluated according to an
illustrative example.
