Juan F. Jiménez | Searcharxiv

arXiv2026

Directional Mollification for Knot-Preserving $C^{\infty}$ Smoothing of Polygonal Chains with Explicit Curvature Bounds

We introduce the \textit{directional mollification} operator, which acts on polygonal chains to produce $C^{\infty}$ curve approximants that are arbitrarily close to the original curve -- pointwise and uniformly on compact subsets -- while strictly interpolating its vertices. Unlike standard mollification, which fails to preserve vertices, this directional construction enables local, vertex-preserving smoothing; modifying a single segment alters the $C^{\infty}$ output only within an explicitly controllable neighborhood. The operator admits closed-form curvature bounds and provides analytic control over curve geometry. Furthermore, we develop a parametric family of smoothing operators that unifies conventional and directional mollification into a single framework. Combining computational simplicity with strict waypoint fidelity, this method is directly applicable to geometric modeling, robotics, computer graphics, and CNC machining.

cs.RO↗

arXiv2026

Efficient Path Generation with Curvature Guarantees by Mollification

Path generation, the process of converting high-level mission specifications, such as sequences of waypoints from a path planner, into smooth, executable paths, is a fundamental challenge in mobile robotics. Most path following and trajectory tracking algorithms require the desired path 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, path generation methods must bridge the gap between convenient but non-differentiable planning outputs, such as piecewise linear segments, and the differentiability requirements imposed by downstream control algorithms. 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 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 analytically shown to be computationally more efficient than standard interpolation methods, enabling real-time implementation on microcontrollers, while remaining compatible with standard trajectory tracking and path following algorithms.

cs.RO↗