Geometry and Topology of Polygonal Linkages

Connelly, Robert; Demaine, Erik D.

doi:10.1201/9781420035315.ch9

Cited by 16 publications

(22 citation statements)

References 34 publications

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“…the links which drive A could move to arbitrary poses. But the problem of determining the reachability of arbitrary configurations of A from a starting configuration is known to be PSPACE-hard even for the restricted case of 2D revolute linkages [6,7].…”

Section: Structure Abstractionmentioning

confidence: 99%

Hierarchical Decomposition and Kinematic Abstraction with Virtual Articulations

Vona¹

2010

Advances in Robot Kinematics: Motion in Man and Machine

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Abstract. We introduce a novel hierarchical model to partition a kinematic system into a set of nested subsystems. This is framed in a mixed real/virtual context, where some joints and links may exist in simulation only. We then use this capability to build a precise form of kinematic abstraction, where a potentially complex subsystem can be virtually replaced by a simpler "interface." Hierarchy and abstraction are interesting because they can help manage complexity in large (100+ DoF) mixed real/virtual mechanisms. We prove that checking if an abstraction is proper is PSPACE-hard, but show that even improper abstractions can be useful. Topological algorithms are presented for decomposing a hierarchical or abstracted kinematic system into subsystems that can be treated in isolation, thus speeding up kinematic computations. We demonstrate on a simulation of a hybrid serial/parallel modular tower with over 100 revolute joints.

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Section: Structure Abstractionmentioning

confidence: 99%

Hierarchical Decomposition and Kinematic Abstraction with Virtual Articulations

Vona¹

2010

Advances in Robot Kinematics: Motion in Man and Machine

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“…If there is a configuration of the linkage, we call the linkage realizable; our terminology is based on the exposition in [10]; a more detailed treatment can be found in [11]. We discuss two computational problems related to linkages: realizability, in Section 3.1 and rigidity, in Section 3.2.…”

Section: Realizability Of Linkagesmentioning

confidence: 99%

“…The reduction from STRETCHABILITY to graph realizability in the proof of Theorem 2.1 is geometric in the following sense: each realization of the graph with length constraints constructed from the pseudo-line arrangement 10 Erdős, Harary, and Tutte [16] defined the dimension of a graph using (non-induced) subgraphs of E n . Later, Erdős and Simonovits [15] introduced Euclidean dimension under the name faithful dimension.…”

Section: Issues Of Precisionmentioning

confidence: 99%

“…9 Figure 1 7 For a discussion of graph drawing assumption, see [48]. 8 Our distinction between realizability of graphs and linkages is not universal, but not unusual; for linkages, [10] is a good reference; for graph realizability, definitions and terminology vary, e.g. realizable graphs are sometimes called Euclidean graphs.…”

Section: Graph Realizabilitymentioning

confidence: 99%

“…The Euclidean dimension of a graph G is the smallest n so that G is a (induced) subgraph of E n , a notion introduced by Erdős, Harary, and Tutte [16]. 10 …”

Section: Euclidean Dimension and Unit Distance Graphsmentioning

confidence: 99%

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Realizability of Graphs and Linkages

Schaefer

2012

Thirty Essays on Geometric Graph Theory

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We show that deciding whether a graph with given edge lengths can be realized by a straight-line drawing has the same complexity as deciding the truth of sentences in the existential theory of the real numbers, ETR; we introduce the class ∃R that captures the computational complexity of ETR and many other problems. The graph realizability problem remains ∃R-complete if all edges have unit length, which implies that recognizing unit distance graphs is ∃R-complete. We also consider the problem for linkages: in a realization of a linkage vertices are allowed to overlap and lie on the interior of edges. Linkage realizability is ∃R-complete and remains so if all edges have unit length. A linkage is called rigid if any slight perturbation of its vertices which does not break the linkage (i.e. keeps edge-lengths the same) is the result of a rigid motion of the plane. Testing whether a configuration is not rigid is ∃R-complete.

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Realization of Simply Connected Polygonal Linkages and Recognition of Unit Disk Contact Trees

Bowen

Durocher

Löffler

et al. 2015

Lecture Notes in Computer Science

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We wish to decide whether a simply connected flexible polygonal structure can be realized in Euclidean space. Two models are considered: polygonal linkages (body-and-joint framework) and contact graphs of unit disks in the plane.(1) We show that it is strongly NP-hard to decide whether a given polygonal linkage is realizable in the plane when the bodies are convex polygons and their contact graph is a tree; the problem is weakly NP-hard already for a chain of rectangles, but efficiently decidable for a chain of triangles hinged at distinct vertices. (2) We also show that it is strongly NP-hard to decide whether a given tree is the contact graph of interior-disjoint unit disks in the plane. P and the hinges in H, and edges represent the polygon-hinge incidences.

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Geometry and Topology of Polygonal Linkages

Cited by 16 publications

References 34 publications

Hierarchical Decomposition and Kinematic Abstraction with Virtual Articulations

Hierarchical Decomposition and Kinematic Abstraction with Virtual Articulations

Realizability of Graphs and Linkages

Realization of Simply Connected Polygonal Linkages and Recognition of Unit Disk Contact Trees

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