Optimal Sampling-Based Motion Planning under Differential Constraints: the Drift Case with Linear Affine Dynamics

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

Optimal Sampling-Based Motion Planning under Differential Constraints: the Driftless Case

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

Swarm Relays: Distributed Self-Healing Ground-and-Air Connectivity Chains

The coordination of robot swarms - large decentralized teams of robots - generally relies on robust and efficient inter-robot communication. Maintaining communication between robots is particularly challenging in field deployments. Unstructured environments, limited computational resources, low bandwidth, and robot failures all contribute to the complexity of connectivity maintenance. In this paper, we propose a novel lightweight algorithm to navigate a group of robots in complex environments while maintaining connectivity by building a chain of robots. The algorithm is robust to single robot failures and can heal broken communication links. The algorithm works in 3D environments: when a region is unreachable by wheeled robots, the chain is extended with flying robots. We test the performance of the algorithm using up to 100 robots in a physics-based simulator with three mazes and different robot failure scenarios. We then validate the algorithm with physical platforms: 7 wheeled robots and 6 flying ones, in homogeneous and heterogeneous scenarios.Comment: 9 pages, 8 figures, Accepted for publication in Robotics and Automation Letters (RAL