Path Planning for Minimizing the Expected Cost until Success

TL;DR

This paper addresses a complex path planning problem aiming to minimize expected costs until success by formulating it as an MDP, proving NP-hardness, and proposing near-optimal and fast heuristic algorithms with extensive real-world scenario evaluations.

Contribution

It introduces a formal framework for the problem, proves its NP-hardness, and develops a game-theoretic and heuristic algorithms that outperform existing methods.

Findings

Game-theoretic planner approaches optimality asymptotically.

Proposed heuristics are fast and non-myopic.

Significant performance gains demonstrated in simulations.

Abstract

Consider a general path planning problem of a robot on a graph with edge costs, and where each node has a Boolean value of success or failure (with respect to some task) with a given probability. The objective is to plan a path for the robot on the graph that minimizes the expected cost until success. In this paper, it is our goal to bring a foundational understanding to this problem. We start by showing how this problem can be optimally solved by formulating it as an infinite horizon Markov Decision Process, but with an exponential space complexity. We then formally prove its NP-hardness. To address the space complexity, we then propose a path planner, using a game-theoretic framework, that asymptotically gets arbitrarily close to the optimal solution. Moreover, we also propose two fast and non-myopic path planners. To show the performance of our framework, we do extensive simulations for two scenarios: a rover on Mars searching for an object for scientific studies, and a robot looking for a connected spot to a remote station (with real data from downtown San Francisco). Our numerical results show a considerable performance improvement over existing state-of-the-art approaches.