Explicit Formulas for the Inversion of the Convolution of Polynomials and Arbitrary Functions with Schwartz Kernels

Convolution serves as a powerful operation for the regularization of functions. While polynomials inherently possess smoothness, it is particularly interesting to investigate their behavior under convolution. This interest stems from the fact that numerous engineering and physical phenomena can be modeled through such operations, including weighted averages, blurring effects and convolutional integral equations. In this work, we show that under certain mild conditions, the convolution with any even Schwartz function acts as an automorphism on the vector space of finite-order polynomials. We derive explicit equations for the inverse operation of this convolution, which are numerically simple to implement. In addition, we extend the deconvolution with (not necessarily even) Schwartz functions to a broader class of functions, including $L^1(\mathbb{R})$, $L^2(\mathbb{R})$, the Schwartz space and tempered distributions. Specifically, we establish a explicit rigorous formula for the deconvolution of a function or distribution that has been convolved with a Schwartz function, being a particular example the Weierstrass Transform. For the latter, we show that any Schwartz function and tempered distribution that has been transformed, can be recovered, in their respective topologies, by the limit of a sequence of linear combination of recursive convolutions. This provides a new formula for the inverse of the Weierstrass Transform that can be numerically implemented.

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.

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.