Directional Mollification for Controlled Smooth Path Generation

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.

Efficient Generation of Smooth Paths with Curvature Guarantees by Mollification

Most path following and trajectory tracking algorithms in mobile robotics require the desired path or trajectory to be defined by at least twice continuously differentiable functions to guarantee key properties such as global convergence, especially for nonholonomic robots like unicycles with speed constraints. Consequently, these algorithms typically exclude continuous but non-differentiable paths, such as piecewise functions. Despite this exclusion, such paths provide convenient high-level inputs for describing robot missions or behavior. While techniques such as spline interpolation or optimization-based methods are commonly used to smooth non-differentiable paths or create feasible ones from sequences of waypoints, they either can produce unnecessarily complex trajectories or are computationally expensive. In this work, we present a method to regularize non-differentiable functions and generate feasible paths through mollification. Specifically, we approximate an arbitrary path with a differentiable function that can converge to it with arbitrary precision. Additionally, we provide a systematic method for bounding the curvature of generated paths, which we demonstrate by applying it to paths resulting from linking a sequence of waypoints with segments. The proposed approach is computationally efficient, enabling real-time implementation on microcontrollers and compatibility with standard trajectory tracking and path following algorithms.