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Author Reif, John ♦ Sharir, Micha
Source ACM Digital Library
Content type Text
Publisher Association for Computing Machinery (ACM)
File Format PDF
Copyright Year ©1994
Language English
Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science
Subject Keyword Turing machines ♦ Computational geometry ♦ Cylindrical algebraic decomposition ♦ Decision procedures ♦ Motion planning ♦ Moving obstacles ♦ Theory of reals
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.
ISSN 00045411
Age Range 18 to 22 years ♦ above 22 year
Educational Use Research
Education Level UG and PG
Learning Resource Type Article
Publisher Date 1994-07-01
Publisher Place New York
e-ISSN 1557735X
Journal Journal of the ACM (JACM)
Volume Number 41
Issue Number 4
Page Count 27
Starting Page 764
Ending Page 790

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Source: ACM Digital Library