Probabilistic completeness is an important property in motion planning.
Although it has been established with clear assumptions for geometric planners,
the panorama of completeness results for kinodynamic planners is still
incomplete, as most existing proofs rely on strong assumptions that are
difficult, if not impossible, to verify on practical systems. In this paper, we
focus on an important class of kinodynamic planners, namely those that
interpolate trajectories in the state space. We provide a proof of
probabilistic completeness for these planners under assumptions that can be
readily verified from the system's equations of motion and the user-defined
interpolation function. Our proof relies crucially on a property of
interpolated trajectories, termed second-order continuity (SOC), which we show
is tightly related to the ability of a planner to benefit from denser sampling.
We analyze the impact of this property in simulations on a low-torque pendulum.
Our results show that a simple RRT using a second-order continuous
interpolation swiftly finds solution, while it is impossible for the same
planner using standard Bezier curves (which are not SOC) to find any solution.Comment: 21 pages, 5 figure