Fastest path across constrained moving rectilinear obstacles
TL;DR: The fastest path across constrained moving rectilinear obstacles is shown, and the optimal path is described, for moving parallel obstacles.
About: This article is published in Information Processing Letters.The article was published on 1991-03-28. It has received 1 citations till now. The article focuses on the topics: Motion planning & Computational geometry.
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TL;DR: A bibliography of nearly 1200 references related to computer vision and image analysis, arranged by subject matter is presented, covering topics including architectures; computational techniques; feature detection and segmentation; image analysis; and motion.
Abstract: This paper presents a bibliography of nearly 1200 references related to computer vision and image analysis, arranged by subject matter. The topics covered include architectures; computational techniques; feature detection and segmentation; image analysis; two-dimensional shape; pattern; color and texture; matching and stereo; three-dimensional recovery and analysis; three-dimensional shape; and motion. A few references are also given on related topics, such as geometry, graphics, image input/output and coding, image processing, optical processing, visual perception, neural nets, pattern recognition and artificial intelligence, as well as on applications.
16 citations
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TL;DR: The goal is to find interesting cases for which the solution can be obtained without the explicit construction of the entire visibility graph, which solve the problems by constructing the shortest-path tree from the source to all the vertices of the obstacles and to the destination.
Abstract: In this paper we address the problem of constructing a Euclidean shortest path between two specified points (source, destination) in the plane, which avoids a given set of barriers. This problem had been solved earlier for polygonal obstacles with the aid of the visibility graph. This approach however, has an Ω(n2) time lower bound, if n is the total number of vertices of the obstacles. Our goal is to find interesting cases for which the solution can be obtained without the explicit construction of the entire visibility graph. The two cases are (i) the path must lie within an n-vertex simple polygon; (ii) the obstacles are n disjoint and parallel line segments. In both instances greedy O(n log n) time algorithms can be developed which solve the problems by constructing the shortest-path tree from the source to all the vertices of the obstacles and to the destination.
428 citations
TL;DR: Evidence that the 3-D dynamic movement problem is intractable even if B has only a constant number of degrees of freedom of movement is provided, and evidence that the problem is PSPACE-hard if B is given a velocity modulus bound on its movements.
Abstract: This paper investigates the computational complexity of planning the motion of a body B in 2-D or 3-D space, so as to avoid collision with moving obstacles of known, easily computed, trajectories. Dynamic movement problems are of fundamental importance to robotics, but their computational complexity has not previously been investigated.We provide evidence that the 3-D dynamic movement problem is intractable even if B has only a constant number of degrees of freedom of movement. In particular, we prove the problem is PSPACE-hard if B is given a velocity modulus bound on its movements and is NP-hard even if B has no velocity modulus bound, where, in both cases, B has 6 degrees of freedom. To prove these results, we use a unique method of simulation of a Turing machine that uses time to encode configurations (whereas previous lower bound proofs in robotic motion planning used the system position to encode configurations and so required unbounded number of degrees of freedom).We also investigate a natural class of dynamic problems that we call asteroid avoidance problems: B, the object we wish to move, is a convex polyhedron that is free to move by translation with bounded velocity modulus, and the polyhedral obstacles have known translational trajectories but cannot rotate. This problem has many applications to robot, automobile, and aircraft collision avoidance. Our main positive results are polynomial time algorithms for the 2-D asteroid avoidance problem, where B is a moving polygon and we assume a constant number of obstacles, as well as single exponential time or polynomial space algorithms for the 3-D asteroid avoidance problem, where B is a convex polyhedron and there are arbitrarily many obstacles. Our techniques for solving these asteroid avoidance problems use “normal path” arguments, which are an intereting generalization of techniques previously used to solve static shortest path problems.We also give some additional positive results for various other dynamic movers problems, and in particular give polynomial time algorithms for the case in which B has no velocity bounds and the movements of obstacles are algebraic in space-time.
164 citations
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02 Sep 2011TL;DR: The problem is analyzed, and polynomial-time motion-planning algorithms are given for the case of a particle moving in one dimension.
Abstract: A bodyB must move from a placementZ0 to a placementZ1, while avoiding collision with a setS of moving obstacles. The motion must satisfy an inertial constraint: the acceleration cannot exceed a given boundM. The problem is analyzed, and polynomial-time motion-planning algorithms are given for the case of a particle moving in one dimension.
87 citations
TL;DR: It is shown that a path existence problem in time-dependent graphs is PSPACE-complete and it is demonstrated that a version of the motion planning problem is PSPace-hard, even if D=2, B is a square and the obstacles have only translational movement.
Abstract: In this paper we study the problem of motion planning in the presence of time dependent, i.e. moving, obstacles. More specifically, we will consider the problem: given a bodyB and a collection of moving obstacles inD-dimensional space decide whether there is a continuous, collision-free movement ofB from a given initial position to a target position subject to the condition thatB cannot move any faster than some fixed top-speedc. As a discrete, combinatorial model for the continuous, geometric motion planning problem we introduce time-dependent graphs. It is shown that a path existence problem in time-dependent graphs is PSPACE-complete. Using this result we will demonstrate that a version of the motion planning problem (where the obstacles are allowed to move periodically) is PSPACE-hard, even ifD=2, B is a square and the obstacles have only translational movement. ForD=1 it is shown that motion planning is NP-hard. Furthermore we introduce the notion of thec-hull of an obstacle: thec-hull is the collection of all positions in space-time at which a future collision with an obstacle cannot be avoided. In the low-dimensional situationD=1 andD=2 we develop polynomial-time algorithms for the computation of thec-hull as well as for the motion planning problem in the special case where the obstacles are polyhedral. In particular forD=1 there is aO(n lgn) time algorithm for the motion planning problem wheren is the number of edges of the obstacle.
46 citations
TL;DR: The problem is reduced to finding the shortest path between two points A and B in an approximate network with vertex set V = W ∪ {A, B}.
Abstract: Given a disjoint planar set, {Li, = PiQi, i = 1,…, n}, of line segments called barriers, we consider the question of finding a path Γ of minimal length which connects two given points A and B and which does not “cut” any of the barriers. In Section III we show that such a minimal path exists and that it is polygonal with its bend points lying in W = {Pi, Qi: i = 1,…, n}. The problem is thus reduced to finding the shortest path between two points A and B in an approximate network with vertex set V = W ∪ {A, B}. The latter can be solved by a network routing algorithm such as Dantzig's. Section IV presents Algorithms for reducing the size of the network.
5 citations