Optimal Sampling-Based Motion Planning under Differential Constraints: the Drift Case with Linear Affine Dynamics
This work addresses motion planning challenges for systems with momentum, such as robotics, offering a foundational step for extending to non-linear systems, though it is incremental as it builds on existing sampling-based methods.
The paper tackles the problem of optimal motion planning for drift control systems with linear affine dynamics, providing a theoretical framework and an asymptotically optimal sampling-based algorithm (Differential Fast Marching Tree) with convergence guarantees and concrete bounds on the convergence rate.
In this paper we provide a thorough, rigorous theoretical framework to assess optimality guarantees of sampling-based algorithms for drift control systems: systems that, loosely speaking, can not stop instantaneously due to momentum. We exploit this framework to design and analyze a sampling-based algorithm (the Differential Fast Marching Tree algorithm) that is asymptotically optimal, that is, it is guaranteed to converge, as the number of samples increases, to an optimal solution. In addition, our approach allows us to provide concrete bounds on the rate of this convergence. The focus of this paper is on mixed time/control energy cost functions and on linear affine dynamical systems, which encompass a range of models of interest to applications (e.g., double-integrators) and represent a necessary step to design, via successive linearization, sampling-based and provably-correct algorithms for non-linear drift control systems. Our analysis relies on an original perturbation analysis for two-point boundary value problems, which could be of independent interest.