A geometric investigation of reach
A geometric investigation of reach
Analysis of Mechanisms and Robot Manipulators
Analysis of Mechanisms and Robot Manipulators
Acute Triangulations of Polygons
Discrete & Computational Geometry
Fundamentals of Robotic Mechanical Systems: Theory, Methods, and Algorithms (Mechanical Engineering Series)
On the workspace boundary determination of serial manipulators with non-unilateral constraints
Robotics and Computer-Integrated Manufacturing
A theorem on polygon cutting with applications
SFCS '82 Proceedings of the 23rd Annual Symposium on Foundations of Computer Science
Flattening single-vertex origami: the non-expansive case
Proceedings of the twenty-fifth annual symposium on Computational geometry
Single-Vertex origami and spherical expansive motions
JCDCG'04 Proceedings of the 2004 Japanese conference on Discrete and Computational Geometry
Extremal reaches in polynomial time
Proceedings of the twenty-seventh annual symposium on Computational geometry
Exact workspace boundary by extremal reaches
Proceedings of the twenty-seventh annual symposium on Computational geometry
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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.