A regularization method for planar offset curves and bi-offset recognition

TL;DR

This paper introduces a novel regularization and approximation method for planar offset curves, improving their practical usability in industrial and autonomous systems by reducing singularities and ensuring smoothness.

Contribution

The authors propose a new approach combining penalized Hermite spline regression with geometric considerations to produce well-behaved offset and bi-offset curves.

Findings

Numerical experiments demonstrate the effectiveness of the proposed method.

The approach mitigates singularities and self-intersections in offset curves.

The method enhances the reliability of trajectory planning applications.

Abstract

Offset curves for planar trajectories are interesting in the generation of tool paths for numerically controlled industrial machines and in trajectory planning methods for autonomous driving systems. Theoretical offset curves may exhibit peculiar singularities, including self-intersections, which limit their use in practical applications. Existing approaches address these issue through geometric filtering techniques to detect and remove undesirable features but the computation of accurate and well-behaved offset curves remains a challenging task. We assume a first stage of functional approximation of trajectories by penalized Hermite spline regression enabling the simultaneous fitting of positions and tangents. The regularization is imposed on the second derivatives, effectively mitigating the jerk effect, which is particularly relevant in motion planning and path smoothing applications. Then, taking into account the geometrical pointwise properties of the resulting curve, we design two offset curves through the simultaneous approximation of function values and derivatives. Then, a mathematical model to obtain the so-called bi-offset as most fitting as with the original generator curve is proposed, also relating the offset range and pointwise curvature values. The adaptive reconstruction of the center line from the external boundaries is a topic of interest and is the main focus of our work. Numerical experiments confirm the reliability of our approach at every stage of the resolution process.