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We consider a random variable expressed as the Euclidean distance between an arbitrary point and a random variable uniformly distributed in a closed and bounded set of a three-dimensional Euclidean space. Four cases are considered for this set: a union of disjoint disks, a union of disjoint balls, a union of disjoint line segments, and the boundary of a polyhedron. In the first three cases, we provide closed-form expressions of the cumulative distribution function and the density. In the last case, we propose two algorithms with complexity O (n ln n) , n being the number of edges of the polyhedron, that computes exactly the cumulative distribution function. An application of these results to probabilistic seismic hazard analysis and extensions are discussed. Finally, we present an open source library, available at https://github.com/vguigues/Areas%5fLibrary , that implements the algorithms presented in this paper. [ABSTRACT FROM AUTHOR]

Exploration algorithms based on the Boustrophedon path seldom consider the impacts of a robot turning at corners on the exploration time. This paper proposes the Forecast-Island and Bidding A*-Euclidean Selecting Boustrophedon Coordination (FIBA*ESBC) algorithm to calculate the turning time at corners in the overall exploration time and introduces a method to estimate the walking time in the Boustrophedon paths in order to determine the directions for path execution. Typically, in bidding-based exploration tasks, the cost is the Euclidean distance between the current position of the robot and the target point. When there is an obstacle between two points, the cost is set to infinity. Therefore, the selected target point is sometimes not optimal. The FIBA*ESBC algorithm is based on the exploration cost of a combination of the Euclidean distance and A* algorithm walking path, which can effectively solve this problem. Because the bidding is based on a greedy algorithm, the robot has a small unexplored island in the later exploration stage; therefore, full exploration is not possible or requires a long time with several repeated paths. The FIBA*ESBC algorithm prioritizes the exploration and estimation of hidden and existing unexplored islands. It can realize complete exploration and decrease the exploration time. Through simulation experiments conducted using Gazebo and RViz, the feasibility of the FIBA*ESBC algorithm is verified. Moreover, a simulation experiment is conducted in MATLAB for comparison with other algorithms. The analysis of the experimental data shows that the proposed algorithm has a relatively short exploration time. [ABSTRACT FROM AUTHOR]