Computing Smooth Geodesics under Two-Sided Curvature Bounds with Applications to Robotics and Image Analysis
This work addresses the challenge of enforcing two-sided curvature constraints in path planning, which is important for robotics and image analysis applications requiring smooth, bounded-curvature paths.
The authors propose a curvature-bounded geodesic model under the HJB PDE framework to compute minimal paths with arbitrary upper and lower curvature bounds, and demonstrate its effectiveness in robot path planning and image analysis tasks.
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.