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Singularities of Nonredundant Manipulators: A Short Account and a Method for their Computation in the Planar Case
, 2013
"... The study of the singularity set is of utmost utility in understanding the local and global behavior of a manipulator. After reviewing the mathematical conditions that characterize this set, and their significance, this paper shows how these conditions can be formulated in an amenable manner in the ..."
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The study of the singularity set is of utmost utility in understanding the local and global behavior of a manipulator. After reviewing the mathematical conditions that characterize this set, and their significance, this paper shows how these conditions can be formulated in an amenable manner in the planar case, allowing to define a conceptuallysimple method for isolating the set exhaustively, even in higherdimensional cases. As a result, the method delivers a collection of boxes bounding the location of all points of the set, whose accuracy can be adjusted through a threshold parameter. Such boxes can then be projected to the input or output coordinate spaces, obtaining informative diagrams, or portraits, on the global motion capabilities of the manipulator. Examples are included that show the application of the method to simple manipulators, and to a complex mechanism that would be difficult to analyze using commonpractice procedures.
Symmetric Laman theorems for the groups C2 and Cs
"... For a bar and joint framework (G,p) with point group C3 which describes 3fold rotational symmetry in the plane, it was recently shown in (Schulze, Discret. Comp. Geom. 44:946972) that the standard Laman conditions, together with the condition derived in (Connelly et al., Int. J. Solids Struct. 46: ..."
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For a bar and joint framework (G,p) with point group C3 which describes 3fold rotational symmetry in the plane, it was recently shown in (Schulze, Discret. Comp. Geom. 44:946972) that the standard Laman conditions, together with the condition derived in (Connelly et al., Int. J. Solids Struct. 46:762773) that no vertices are fixed by the automorphism corresponding to the 3fold rotation (geometrically, no vertices are placed on the center of rotation), are both necessary and sufficient for (G,p) to be isostatic, provided that its joints are positioned as generically as possible subject to the given symmetry constraints. In this paper we prove the analogous Lamantype conjectures for the groups C2 and Cs which are generated by a halfturn and a reflection in the plane, respectively. In addition, analogously to the results in (Schulze, Discret. Comp. Geom. 44:946972), we also characterize symmetry generic isostatic graphs for the groups C2 and Cs in terms of inductive Hennebergtype constructions, as well as Crapotype 3Tree2 partitions the full sweep of methods used for the simpler problem without symmetry. 1
Geometric Properties of Assur Graphs
, 2008
"... In our previous paper, we presented the combinatorial theory for minimal isostatic pinned frameworks Assur graphs which arise in the analysis of mechanical linkages. In this paper we further explore the geometric properties of Assur graphs, with a focus on singular realizations which have static s ..."
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In our previous paper, we presented the combinatorial theory for minimal isostatic pinned frameworks Assur graphs which arise in the analysis of mechanical linkages. In this paper we further explore the geometric properties of Assur graphs, with a focus on singular realizations which have static selfstresses. We provide a new geometric characterization of Assur graphs, based on special singular realizations. These singular positions are then related to deadend positions in which an associated mechanism with an inserted driver will stop or jam. 1
Rootedtree decompositions with matroid constraints and infinitesimal rigidity of frameworks with boundaries
, 2011
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Directed graphs, decompositions, and spatial linkages
 Discret. Appl. Math. 2013
"... The decomposition of a linkage into minimal components is a central tool of analysis and synthesis of linkages. In this paper we prove that every pinned disostatic (minimally rigid) graph (grounded linkage) has a unique decomposition into minimal strongly connected components (in the sense of direc ..."
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The decomposition of a linkage into minimal components is a central tool of analysis and synthesis of linkages. In this paper we prove that every pinned disostatic (minimally rigid) graph (grounded linkage) has a unique decomposition into minimal strongly connected components (in the sense of directed graphs), or equivalently into minimal pinned isostatic graphs, which we call dAssur graphs. We also study key properties of motions induced by removing an edge in a dAssur graph defining a stronger subclass of strongly dAssur graphs by the property that all inner vertices go into motion, for each removed edge. The strongly 3Assur graphs are the central building blocks for kinematic linkages in 3space and the 3Assur graphs are components in the analysis of built linkages. The dAssur graphs share a number of key combinatorial and geometric properties with the 2Assur graphs, including an associated lower blocktriangular decomposition of the pinned rigidity matrix which provides modular information for extending the motion induced by inserting one driver in a bottom Assur linkage to the joints of the entire linkage. We also highlight some problems in combinatorial rigidity in higher dimensions (d ≥ 3) which cause the distinction between dAssur and strongly dAssur which did not occur in the plane. ∗Supported by a grant from ISF (Israel)