Planning under Uncertainty to Goal Distributions

Goal spaces for planning problems are typically conceived of as subsets of the state space. It is common to select a particular goal state to plan to, and the agent monitors its progress to the goal with a distance function defined over the state space. Due to numerical imprecision, state uncertainty, and stochastic dynamics, the agent will be unable to arrive at a particular state in a verifiable manner. It is therefore common to consider a goal achieved if the agent reaches a state within a small distance threshold to the goal. This approximation fails to explicitly account for the agent's state uncertainty. Point-based goals further do not accommodate goal uncertainty that arises when goals are estimated in a data-driven way. We argue that goal distributions are a more appropriate goal representation and present a novel approach to planning under uncertainty to goal distributions. We use the unscented transform to propagate state uncertainty under stochastic dynamics and use cross-entropy method to minimize the Kullback-Leibler divergence between the current state distribution and the goal distribution. We derive reductions of our cost function to commonly used goal-reaching costs such as weighted Euclidean distance, goal set indicators, chance-constrained goal sets, and maximum expectation of reaching a goal point. We explore different combinations of goal distributions, planner distributions, and divergence to illustrate behaviors achievable in our framework.