Completeness of Randomized Kinodynamic Planners with State-based Steering
This addresses the challenge of ensuring completeness in motion planning for robotics, offering a more practical verification method, though it is incremental in extending existing geometric results to kinodynamic systems.
The paper tackled the problem of probabilistic completeness for kinodynamic planners that interpolate trajectories in state space, providing a proof under verifiable assumptions and showing in simulations that a simple RRT with second-order continuous interpolation finds solutions quickly, while it fails with standard Bezier curves.
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.