arXiv subjects

Alfredo González-Calvin

Publications and source records attributed to Alfredo González-Calvin.

4 recordsLinked to original sources

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

Exact Deconvolution for Schwartz Kernels: From Polynomial Automorphisms to Recursive Inversion in Tempered Distributions

In this work, we construct an explicit, theoretically rigorous deconvolution method that relies entirely on iterative forward convolutions, thus can be numerically implemented. We first prove that convolution with an even Schwartz kernel acts as an automorphism on the vector space of finite-degree polynomials. Exploiting the parity of the kernel, we derive an exact algebraic inverse for this space, expressed uniquely as a finite linear combination of repeated convolutions. The core contribution of this work extends this algebraic inversion to infinite-dimensional function spaces, including $L^1(\mathbb{R})$, $L^2(\mathbb{R})$, the Schwartz space $\mathscr{S}(\mathbb{R})$, and the space of tempered distributions $\mathscr{S}'(\mathbb{R})$. By passing the finite-sum polynomial inversion formula to the limit, we demonstrate that an arbitrary function or distribution convolved with a Schwartz kernel can be exactly recovered in its respective topology. The resulting inverse is an explicitly computable limit of a sequence of linear combinations of recursive convolutions. As a primary application, this limit provides a fundamentally new, iterative numerical formula for the inverse of the Weierstrass Transform. By bypassing traditional numerically ill-posed inversion techniques, our method offers a mathematically exact and numerically robust algorithm for computational signal recovery.

eess.SP↗

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↗

arXiv2026

Singularity-Free Guiding Vector Field over Bézier's Curves Applied to Rovers Path Planning and Path Following

This paper presents a guidance algorithm for solving the problem of following parametric paths, as well as a curvature-varying speed setpoint for land-based car-type wheeled mobile robots (WMRs). The guidance algorithm relies on Singularity-Free Guiding Vector Fields SF-GVF. This novel GVF approach expands the desired robot path and the Guiding vector field to a higher dimensional space, in which an angular control function can be found to ensure global asymptotic convergence to the desired parametric path while avoiding field singularities. In SF-GVF, paths should follow a parametric definition. This feature makes using Bezier's curves attractive to define the robot's desired patch. The curvature-varying speed setpoint, combined with the guidance algorithm, eases the convergence to the path when physical restrictions exist, such as minimal turning radius or maximal lateral acceleration. We provide theoretical results, simulations, and outdoor experiments using a WMR platform assembled with off-the-shelf components.

cs.RO↗