Directional Mollification for Controlled Smooth Path Generation

Abstract: Path generation, the problem of producing smooth, executable paths from discrete planning outputs, such as waypoint sequences, is a fundamental step in the control of autonomous robots, industrial robots, and CNC machines, as path following and trajectory tracking controllers impose strict differentiability requirements on their reference inputs to guarantee stability and convergence, particularly for nonholonomic systems. Mollification has been recently proposed as a computationally efficient and analytically tractable tool for path generation, offering formal smoothness and curvature guarantees with advantages over spline interpolation and optimization-based methods. However, this mollification is subject to a fundamental geometric constraint: the smoothed path is confined within the convex hull of the original path, precluding exact waypoint interpolation, even when explicitly required by mission specifications or upstream planners. We introduce directional mollification, a novel operator that resolves this limitation while retaining the analytical tractability of classical mollification. The proposed operator generates infinitely differentiable paths that strictly interpolate prescribed waypoints, converge to the original non-differentiable input with arbitrary precision, and satisfy explicit curvature bounds given by a closed-form expression, addressing the core requirements of path generation for controlled autonomous systems. We further establish a parametric family of path generation operators that contains both classical and directional mollification as special cases, providing a unifying theoretical framework for the systematic generation of smooth, feasible paths from non-differentiable planning outputs.