Completeness of Randomized Kinodynamic Planners with State-based Steering

TL;DR

This paper proves probabilistic completeness for a class of kinodynamic planners that interpolate in state space, demonstrating that second-order continuity of trajectories is crucial for planner success, with empirical validation on a pendulum system.

Contribution

It introduces a verifiable proof of probabilistic completeness for state-based interpolating kinodynamic planners, emphasizing the importance of second-order continuity.

Findings

Second-order continuity (SOC) is key for planner success.

A simple RRT with SOC interpolations quickly finds solutions.

Standard Bezier curves without SOC fail to find solutions.

Abstract

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.