Motivated by applications in which a nonholonomic robotic vehicle should sequentially hit a series of waypoints in the presence of stochastic drift, we formulate a new version of the Dubins vehicle traveling salesperson problem. In our approach, we first compute the minimum expected time feedback control to hit one waypoint based on the Hamilton-Jacobi-Bellman equation. Next, minimum expected times associated with the control are used to construct a traveling salesperson problem based on a waypoint hitting angle discretization. We provide numerical results illustrating our solution and analyze how the stochastic drift affects the solution.

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