Core Concepts
This paper presents an efficient and scalable partial-differential-equation-based optimal path-planning framework for curvature-constrained motion in 2D and 3D spaces, with applications to vehicles such as simple cars, airplanes, and submarines.
Abstract
The paper develops a Hamilton-Jacobi partial differential equation (PDE) based method for optimal trajectory generation, with a focus on curvature-constrained motion models known as Dubins vehicles. The authors derive the Hamilton-Jacobi-Bellman (HJB) equations for Dubins car, Dubins airplane, and Dubins submarine models, and propose efficient numerical methods to solve these high-dimensional HJB equations.
Key highlights:
The authors use a level-set formulation to model the optimal path-planning problem, which avoids the need for boundary conditions and allows for efficient grid-free numerical methods.
They develop a saddle-point optimization approach based on the Hopf-Lax formula to approximate the solutions to the HJB equations, which is shown to be efficient and scalable even in high-dimensional state spaces.
The numerical methods maintain the interpretability of the PDE-based approach, while achieving real-time applicability, in contrast to many sampling-based or learning-based motion planning algorithms.
The authors demonstrate their method with several examples of Dubins-type vehicles navigating 2D and 3D environments with obstacles.