Chance-Constrained Motion Planning using Modeled Distance- to-Collision Functions

Time-Optimal Paths for Simple Cars with Moving Obstacles in the Hamilton-Jacobi Formulation

“…Algorithm 1 Splitting Method for Solving (19) Given a point (x, t) ∈ R d × (0, t), a Hamiltonian H, an initial data function g, a time-discretization count N , a max iteration count K, an error tolerance TOL, and relaxation parameters σ, τ, κ > 0, we resolve the minimization problem (19) as follows.…”

“…. , p N } sampled at times 0 = t 0 < t 1 < • • • < t N = t. The integral in (19) is then approximated by a Riemann sum along this path, and the path constraints are enforced discretely. One then alternately solves the minimization problem with respect to the variables p j and x j , includes some relaxation, and iterates until convergence.…”

“…Trajectory generation methods which are rooted in PDE have the advantage that they are easy to implement, entirely interpretable, and can provide theoretical guarantees regarding optimality, robustness, and other concerns. This is to distinguish them from sampling and learning based algorithms (for example [15,16,17,18,19]) which often sacrifice interpretability for efficiency. The main drawbacks of the PDEbased methods are the lack of efficiency and scalability.…”