Optimal Sampling-Based Motion Planning under Differential Constraints: the Driftless Case
This work addresses the lack of optimality guarantees in sampling-based motion planning for robotics, offering a foundational contribution with potential broad impact in the field.
The paper tackles the problem of motion planning under differential constraints by providing a theoretical framework to assess optimality guarantees for sampling-based algorithms, and introduces two novel algorithms that are proven to converge to an optimal solution with convergence rate bounds, supported by numerical experiments.
Motion planning under differential constraints is a classic problem in robotics. To date, the state of the art is represented by sampling-based techniques, with the Rapidly-exploring Random Tree algorithm as a leading example. Yet, the problem is still open in many aspects, including guarantees on the quality of the obtained solution. In this paper we provide a thorough theoretical framework to assess optimality guarantees of sampling-based algorithms for planning under differential constraints. We exploit this framework to design and analyze two novel sampling-based algorithms that are guaranteed to converge, as the number of samples increases, to an optimal solution (namely, the Differential Probabilistic RoadMap algorithm and the Differential Fast Marching Tree algorithm). Our focus is on driftless control-affine dynamical models, which accurately model a large class of robotic systems. In this paper we use the notion of convergence in probability (as opposed to convergence almost surely): the extra mathematical flexibility of this approach yields convergence rate bounds - a first in the field of optimal sampling-based motion planning under differential constraints. Numerical experiments corroborating our theoretical results are presented and discussed.