Filter-Based Abstractions with Correctness Guarantees for Planning under Uncertainty

We study planning problems for continuous control systems with uncertainty caused by measurement and process noise. The goal is to find an optimal plan that guarantees that the system reaches a desired goal state within finite time. Measurement noise causes limited observability of system states, and process noise causes uncertainty in the outcome of a given plan. These factors render the problem undecidable in general. Our key contribution is a novel abstraction scheme that employs Kalman filtering as a state estimator to obtain a finite-state model, which we formalize as a Markov decision process (MDP). For this MDP, we employ state-of-the-art model checking techniques to efficiently compute plans that maximize the probability of reaching goal states. Moreover, we account for numerical imprecision in computing the abstraction by extending the MDP with intervals of probabilities as a more robust model. We show the correctness of the abstraction and provide several optimizations that aim to balance the quality of the plan and the scalability of the approach. We demonstrate that our method can handle systems that result in MDPs with thousands of states and millions of transitions.