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1. Shortest Paths in the Plane with Obstacle Violations

   https://doi.org/s00453-020-00673-y  

   Hershberger, J. ; Kumar, N. ; Suri, S. ( January 2020 , Algorithmica)

   We study the problem of finding shortest paths in the plane among h convex obstacles, where the path is allowed to pass through (violate) up to k obstacles, for 𝑘≤ℎ. Equivalently, the problem is to find shortest paths that become obstacle-free if k obstacles are removed from the input. Given a fixed source point s, we show how to construct a map, called a shortest k-path map, so that all destinations in the same region of the map have the same combinatorial shortest path passing through at most k obstacles. We prove a tight bound of 𝛩(𝑘𝑛) on the size of this map, and show that it can be computed in 𝑂(𝑘2𝑛log𝑛) time, where n is the total number of obstacle vertices. 

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2. A New Algorithm for Euclidean Shortest Paths in the Plane

   https://doi.org/10.1145/3580475  

   Wang, Haitao ( April 2023 , Journal of the ACM)

   Given a set of pairwise disjoint polygonal obstacles in the plane, finding an obstacle-avoiding Euclidean shortest path between two points is a classical problem in computational geometry and has been studied extensively. Previously, Hershberger and Suri (in SIAM Journal on Computing , 1999) gave an algorithm of O(n log n ) time and O(n log n ) space, where n is the total number of vertices of all obstacles. Recently, by modifying Hershberger and Suri’s algorithm, Wang (in SODA’21) reduced the space to O(n) while the runtime of the algorithm is still O(n log n ). In this article, we present a new algorithm of O(n+h log h ) time and O(n) space, provided that a triangulation of the free space is given, where h is the number of obstacles. The algorithm is better than the previous work when h is relatively small. Our algorithm builds a shortest path map for a source point s so that given any query point t , the shortest path length from s to t can be computed in O (log n ) time and a shortest s - t path can be produced in additional time linear in the number of edges of the path. 

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3. Path Planning for Indoor UAV Using A* and Late Acceptance Hill Climbing Algorithms Utilizing Probabilistic Roadmap

   https://doi.org/10.14419/ijet.v9i4.31033  

   Hopkins, Jacob ; Joy, Forrest ; Sheta, Alaa ; Turabieh, Hamza ; Kar, Dulal ( October 2020 , International Journal of Engineering & Technology)

   null (Ed.)

   The main objective of an unmanned aerial vehicle (UAV) path planning is to generate a flight path that links a start point to an endpoint in an indoor space avoiding obstacles.  Path planning is essential for many real-life applications such as an autonomous car, surveillance mission, farming robots, unmanned aerial vehicles package delivery, space exploration, and many others. To create an optimal path, we need to adopt a specific criterion to minimize the distance the UAV must travel such as the Euclidean distance. In this paper, we provide our initial idea of creating an optimal path for indoor UAV using both A* and the Late Acceptance Hill Climbing (LAHC) algorithms. We are adopting an indoor search environment with various complexity and utilize the Probabilistic Roadmap algorithm (PRM) as a search space for both algorithms. The basic idea following PRM is to generate random sample points in the space and search these points for an optimal path. The developed results show that the LAHC algorithm outperforms the A* algorithm. 

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4. Hardness of Motion Planning with Obstacle Uncertainty in Two Dimensions

   https://doi.org/10.1177/0278364921992787  

   Shimanuki, Luke ; Axelrod, Brian ( September 2021 , The International Journal of Robotics Research)

   We consider the problem of motion planning in the presence of uncertain obstacles, modeled as polytopes with Gaussian-distributed faces (PGDFs). A number of practical algorithms exist for motion planning in the presence of known obstacles by constructing a graph in configuration space, then efficiently searching the graph to find a collision-free path. We show that such an exact algorithm is unlikely to be practical in the domain with uncertain obstacles. In particular, we show that safe 2D motion planning among PGDF obstacles is [Formula: see text]-hard with respect to the number of obstacles, and remains [Formula: see text]-hard after being restricted to a graph. Our reduction is based on a path encoding of MAXQHORNSAT and uses the risk of collision with an obstacle to encode variable assignments and literal satisfactions. This implies that, unlike in the known case, planning under uncertainty is hard, even when given a graph containing the solution. We further show by reduction from [Formula: see text]-SAT that both safe 3D motion planning among PGDF obstacles and the related minimum constraint removal problem remain [Formula: see text]-hard even when restricted to cases where each obstacle overlaps with at most a constant number of other obstacles. 

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5. Inferring Obstacles and Path Validity from Visibility-Constrained Demonstrations

   Knuth, Craig ; Chou, Glen ; Ozay, Necmiye ; Berenson, Dmitry ( January 2020 , Workshop on Algorithmic Foundations of Robotics)

   null (Ed.)

   Many methods in learning from demonstration assume that the demonstrator has knowledge of the full environment. However, in many scenarios, a demonstrator only sees part of the environment and they continuously replan as they gather information. To plan new paths or to reconstruct the environment, we must consider the visibility constraints and replanning process of the demonstrator, which, to our knowledge, has not been done in previous work. We consider the problem of inferring obstacle configurations in a 2D environment from demonstrated paths for a point robot that is capable of seeing in any direction but not through obstacles. Given a set of survey points, which describe where the demonstrator obtains new information, and a candidate path, we con-struct a Constraint Satisfaction Problem (CSP) on a cell decomposition of the environment. We parameterize a set of obstacles corresponding to an assignment from the CSP and sample from the set to find valid environments. We show that there is a probabilistically-complete, yet not entirely tractable, algorithm that can guarantee novel paths in the space are unsafe or possibly safe. We also present an incomplete, but empirically-successful, heuristic-guided algorithm that we apply in our experiments to 1) planning novel paths and 2) recovering a probabilistic representation of the environment. 

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