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The convexity of a set can be generalized to the two weaker notions of reach and $r$-convexity; both describe the regularity of a set's boundary. For any compact subset of $\mathbb{R}^d$, we provide methods for computing upper bounds on these quantities from point cloud data. The bounds converge to the respective quantities as the point cloud becomes dense in the set, and the rate of convergence for the bound on the reach is given under a weak regularity condition. We also introduce the $\beta$-reach, a generalization of the reach that excludes small-scale features of size less than a parameter $\beta\in[0,\infty)$. Numerical studies suggest how the $\beta$-reach can be used in high-dimension to infer the reach and other geometric properties of smooth submanifolds., Comment: 37 pages, 23 figures

Small unmanned aerial vehicles (UAVs) have generally little capacity to carry payloads. Through collaboration, the UAVs can increase their joint payload capacity and carry more significant loads. For maximum flexibility to dynamic and unstructured environments and task demands, we propose a fully decentralized control infrastructure based on a swarmspecific scripting language, Buzz. In this paper, we describe the control infrastructure and use it to compare two algorithms for collaborative transport: field potentials and spring-damper. We test the performance of our approach with a fleet of micro- UAVs, demonstrating the potential of decentralized control for collaborative transport.

Small unmanned aerial vehicles (UAVs) have generally little capacity to carry payloads. Through collaboration, the UAVs can increase their joint payload capacity and carry more significant loads. For maximum flexibility to dynamic and unstructured environments and task demands, we propose a fully decentralized control infrastructure based on a swarm-specific scripting language, Buzz. In this paper, we describe the control infrastructure and use it to compare two algorithms for collaborative transport: field potentials and spring-damper. We test the performance of our approach with a fleet of micro-UAVs, demonstrating the potential of decentralized control for collaborative transport., Comment: Submitted to 2019 International Conference on Robotics and Automation (ICRA). 6 pages