How Far Can You Reach?

Ciprian Borcea, Rider University
Ileana Streinu, Smith College

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The problem of computing the maximum reach configurations of a 3D revolute-jointed manipulator is a long-standing open problem in robotics. In this paper we present an optimal algorithmic solution for orthogonal polygonal chains. This appears as a special case of a larger family, fully characterized here by a technical condition. Until now, in spite of the practical importance of the problem, only numerical optimization heuristics were available, with no guarantee of obtaining the global maximum. In fact, the problem was not even known to be computationally solvable, and in practice, the numerical heuristics were applicable only to small problem sizes. We present elementary and efficient (mostly linear) algorithms for four fundamental problems: (1) finding the maximum reach value, (2) finding a maximum reach configuration (or enumerating all of them), (3) folding a given chain to a given maximum position, and (4) folding a chain in a way that changes the endpoint distance function monotonically. The algorithms rely on our recent theoretical results characterizing combinatorially the maximum of panel-and-hinge chains. They allow us to reduce the first problem to finding a shortest path between two vertices in an associated simple triangulated polygon, and the last problem to a simple version of the planar carpenter's rule problem. Copyright © by SIAM.