Simplicial Label Correcting Algorithms for continuous stochastic shortest path problems

The problem of optimal feedback planning under prediction uncertainties among static obstacles is considered. A discrete-time stochastic state transition model is defined over a continuous state space. A relation to a continuous 'nearby' deterministic model is proven for small time steps; the cost-to-go function of the stochastic model is approximated with that of the deterministic model, and the approximation error is found to be proportional to the time step. This motivates using numerical methods, which are vastly available for solving deterministic problems, to approximate the original stochastic problem. We demonstrate this application using a Simplicial Label Correcting Algorithm. This algorithms uses a piecewise linear discretization to compute the shortest-path plan on a simplicial complex. Additionally, the theoretical error bound between the approximate solution and the exact solution is derived and confirmed during numerical experiments. This paper provides a rigorous analysis as well as algorithmic and implementation details of the proposed model for the stochastic shortest path problem in continuous spaces with obstacles.