Computing Smooth Geodesics under Two-Sided Curvature Bounds with Applications to Robotics and Image Analysis

2026-06-11

AI Summary

This work addresses the problem of smooth path tracing under simultaneous upper and lower curvature bounds by proposing a novel curvature-bounded geodesic model. Formulated within the Hamilton-Jacobi-Bellman (HJB) partial differential equation framework, the model introduces bilateral curvature constraints into the HJB formalism for the first time, enabling strong control over the geometric properties of generated paths. An efficient numerical discretization scheme is devised to balance path smoothness, rigidity, and elasticity. Experimental results demonstrate that the method robustly produces high-quality, curvature-constrained optimal paths in applications such as robotic motion planning and image curve structure tracking, significantly extending the capabilities of conventional single-bound constrained models.

Abstract

Curvature of planar curves serves as a key regularization term for computing second-order minimal paths, due to its tight relevance to desirable geometric properties such as smoothness, rigidity, and elasticity. In this paper, we tackle a more challenging problem in computational physics and geometry problem: tracking minimal paths whose curvature is constrained by arbitrary upper and lower bounds. For that purpose, we propose a new curvature-bounded geodesic model, developed under the Hamilton-Jacobi-Bellman (HJB) partial differential equation (PDE) framework. It provides strong geometric control over minimal paths by enforcing curvature range constraints, whose paths are smooth and of bounded curvature limitation. We also present a discretization scheme for the Hamiltonian and the HJB PDE incorporating curvature bounds, allowing efficient solver for estimating numerical solutions to the model. Finally, we illustrate the capability of the proposed curvature-bounded geodesic model in applications of robot path planning and curvilinear structures tracking from images. Numerical experiments demonstrate that the proposed curvature-bounded geodesic model serves as a powerful and robust tool for finding satisfactory paths.

Problem

• curvature-bounded geodesics
• minimal paths
• two-sided curvature bounds
• smoothness
• path planning

Innovation

• curvature-bounded geodesics
• Hamilton-Jacobi-Bellman PDE
• two-sided curvature constraints
• path planning
• curvilinear structure tracking