The global spread of the COVID-19 pandemic has followed complex pathways, largely attributed to the high virus infectivity, human travel patterns, and the implementation of multiple mitigation measures. The resulting geographic patterns describe the evolution of the epidemic and can indicate areas that are at risk of an outbreak. Here, we analyze the spatial correlations of new active cases in the USA at the county level and characterize the extent of these correlations at different times. We show that the epidemic did not progress uniformly and we identify various stages which are distinguished by significant differences in the correlation length. Our results indicate that the correlation length may be large even during periods when the number of cases declines. We find that correlations between urban centers were much more significant than between rural areas and this finding indicates that long-range spreading was mainly facilitated by travel between cities, especially at the first months of the epidemic. We also show the existence of a percolation transition in November 2020, when the largest part of the country was connected to a spanning cluster, and a smaller-scale transition in January 2021, with both times corresponding to the peak of the epidemic in the country.

Abstract-Probabilistic Roadmap Methods (PRMs) are one of the most used classes of motion planning methods. These sampling-based methods generate robot configurations (nodes) and then connect them to form a graph (roadmap) containing representative feasible pathways. A key step in PRM roadmap construction involves identifying a set of candidate neighbors for each node. Traditionally, these candidates are chosen to be the k-closest nodes based on a given distance metric. In this paper, we propose a new neighbor selection policy called LocalRand(k, k ′ ), that first computes the k ′ closest nodes to a specified node and then selects k of those nodes at random. Intuitively, LocalRand attempts to benefit from random sampling while maintaining the higher levels of local planner success inherent to selecting more local neighbors. We provide a methodology for selecting the parameters k and k ′ . We perform an experimental comparison which shows that for both rigid and articulated robots, LocalRand results in roadmaps that are better connected than the traditional k-closest policy or a purely random neighbor selection policy. The cost required to achieve these results is shown to be comparable to k-closest. I. IntroductionThe motion planning problem involves finding a valid path for a movable object (robot) from a start to a goal configuration in a given environment. Motion planning is an important component of many applications, including computer-aided design Sampling-based methods have been able to solve many motion planning problems that exact methods cannot. One of the most common randomized methods is the Probabilistic Roadmap Method, or PRM [19]. This method randomly generates valid samples (nodes) in an environment's configuration space (C-space) and then attempts to connect nearby pairs of nodes using a local planner, a simplified planner whose objective is to generate and validate transitions between the the specified pair of nodes. The resulting graph, or roadmap, encodes representative feasible paths in C-space and can be queried to obtain valid paths in the environment.One of the key steps in PRM construction is node connection. Ideally, roadmap connectivity should reflect the connectivity of the underlying C-space. From this perspective, the best strategy would be to attempt to connect all θ (n 2 ) pairs of nodes. However, the cost of all these connection attempts is not feasible for any but the simplest of problems. Hence, the selection of candidates for local transitions (neighbors) is crucial to both roadmap quality and efficiency.The objective of a good neighbor selection strategy is to identify pairs of configurations that have a high probability of being connectible by the local planner and that are useful in terms of producing good quality roadmaps. The most commonly used method for neighbor selection in PRMs uses nearest-neighbor search to select the k nodes that are closest to the node in question, where k is typically some relatively small, fixed constant, typically between 5 and 25 [10].

We introduce a new concept, reachable volumes, which denotes the set of points that the end effector of a chain or linkage can reach. This generalizes work on reachable distances for planar robots to 3-dimensional spherical joints. We show that the reachable volume of a chain is equivalent to the Minkowski sum of the reachable volumes of its links, which gives us an efficient method for computing reachable volumes. We present a method for generating configurations using reachable volumes for various types of robots including chain robots (with and without closure constraints), tree-like robots, and robots containing a combination of them. Unlike previous methods, ours works for 3-dimensional linkages with spherical joints and is capable of generating samples for problems with constraints on internal joints as well as end effectors. We show that reachable volumes samples are less likely to be invalid due to self-collisions, making reachable volumes sampling more efficient for higher dimensional problems. We also show that these samples are easier to connect than others, resulting in more connected roadmaps. Finally we demonstrate that our method can be applied to 262-dof, multi-loop, and tree-like linkages, problems where existing methods cannot be used (e.g., closed chains with spherical joints) or cannot be solved efficiently (e.g., tree-like robots and high degree of freedom chains with spherical joints).

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