This paper presents a strategy for stochastic control of small aerial vehicles under uncertainty using graph-based methods. In planning with graph-based methods, such as the probabilistic roadmap method (PRM) in state space or the information roadmaps (IRM) in information-state (belief) space, the local planners (along the edges) are responsible to drive the state/belief to the final node of the edge. However, for aerial vehicles with minimum velocity constraints, driving the system belief to a sampled belief is a challenge. In this paper, we propose a novel method based on periodic controllers, in which instead of stabilizing the belief to a predefined probability distribution, the belief is stabilized to an orbit (periodic path) of probability distributions. Choosing nodes along these orbits, the node reachability in belief space is achieved and we can form a graph in belief space that can handle higher order dynamics or nonstoppable systems (whose velocity cannot be zero), such as fixed-wing aircraft. The proposed method takes obstacles into account and provides a query-independent graph, since its edge costs are independent of each other. Thus, it satisfies the principle of optimality. Therefore, dynamic programming (DP) can be utilized to compute the best feedback on the graph. We demonstrate the method's performance on a unicycle robot and a six degrees of freedom (DoF) small aerial vehicle.

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