Numerical simulation of collision-free near-shortest path generation for Dubins vehicle via Hamilton–Jacobi–Bellman equation: A case study

Abstract

Motivated by modeling an unmanned aerial vehicle at a constant altitude plane above ground level as Dubins vehicle with signed upper-bounded curvature as constrained input and the flight path as Dubins path, the class of flight paths we study is restricted to the paths composed by path segments with zero or constant curvature satisfying the curvature constraint. We start with time-optimal control formulation incorporating mixed constraints for Dubins vehicle in cluttered planar environment and derive the associated Hamilton-Jacobi-Bellman (HJB) equation for the value function from the necessary time-optimality condition. The Kruzkov transform is used to modify the HJB equation for scaling of the solution to be bounded near the obstacles. Numerical solutions based on the fast-sweeping methods with consistent, monotone upwind Godunov Hamiltonian and semi-Lagrange methods are implemented for solving the two-point boundary value problem of modified HJB equation. Comparative validation simulations suggest the appropriateness of both schemes for the numerical collision-free shortest path via Kruzkov transformed HJB equation but with different convergence rate. Two schemes both perform satisfactory path planning, while Godunov Hamiltonian scheme provides more accurate in order of convergence. It is also found by example simulations that the grid resolution and time-step size required for switching structure is finer than that of the convergence criterion.

Keywords

• dubins path
• hjb equation
• godunov hamiltonian scheme
• semi-lagrange scheme
• obstacle avoidance
• unmanned aerial vehicles