Efficient Path Generation with Curvature Guarantees by Mollification

TL;DR

This paper introduces a mollification-based method for generating smooth, curvature-bounded paths from non-differentiable waypoint sequences, enabling real-time implementation in mobile robotics.

Contribution

It presents a novel mollification technique for path regularization that ensures differentiability and curvature bounds, improving efficiency over traditional methods.

Findings

The method approximates arbitrary paths with arbitrary precision.

It provides systematic curvature bounds for generated paths.

The approach is computationally efficient for real-time applications.

Abstract

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.