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Research Papers: Design of Mechanisms and Robotic Systems

Synthesizing Functional Mechanisms From a Link Soup1

[+] Author and Article Information
Pouya Tavousi, Kazem Kazerounian, Horea Ilies

Department of Mechanical Engineering,
University of Connecticut,
Storrs, CT 06269-3139

2Corresponding author.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received July 21, 2015; final manuscript received April 8, 2016; published online May 4, 2016. Assoc. Editor: David Myszka.

J. Mech. Des 138(6), 062303 (May 04, 2016) (13 pages) Paper No: MD-15-1521; doi: 10.1115/1.4033394 History: Received July 21, 2015; Revised April 08, 2016

The synthesis of functional molecular linkages is constrained by difficulties in fabricating nanolinks of arbitrary shapes and sizes. Thus, classical mechanism synthesis methods, which assume the ability to manufacture any designed links, cannot provide a systematic process for assembling such linkages. We propose a new approach to building functional mechanisms with prescribed mobility by using only elements from a predefined “link soup.” First, we enumerate an exhaustive set of topologies, while employing divide-and-conquer algorithms to control the generation and elimination of redundant topologies. Then, we construct the linkage arrangements for each valid topology. Finally, we output a set of feasible geometries through a positional analysis step that minimizes the error associated with closure of the loops in the linkage while avoiding geometric interference. The proposed systematic approach outputs the ATLAS of candidate mechanisms, which can be further processed for downstream applications. The resulting synthesis procedure is the first of its kind that is capable of synthesizing functional linkages with prescribed mobility constructed from a soup of primitive entities.

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Figures

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Fig. 1

The proposed mechanism synthesis procedure

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Fig. 2

Vertices are grouped by their degrees and the graph is decomposed into two sets of subgraphs: the unipartite subgraphs formed within a degree group and the bipartite subgraphs formed across degree groups

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Fig. 3

Different edge distributions among the subgraphs are computed in a divide-and-conquer fashion to specify the coarse adjacency matrix

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Fig. 4

Graph enumeration is equivalent to finding solutions for the fine adjacency matrix. The task is split into generating two sets of subgraphs: the ones that are constructed on the vertices of the unipartite subgraphs, which correspond to square submatrices along the diagonal of the fine adjacency matrix, and the bipartite subgraphs, which correspond to the remaining rectangular submatrices. Note that the summation of elements in the submatrix associated with the intersection of groups i and j reflects a¯ij.

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Fig. 5

The characteristic matrix captures the similarities between vertices or vertex pairs. In the sample update functions,  pcij and  ucij are, respectively, the previous and updated values of the element of row i and column j of the characteristic matrix,  nni is the number neighbors of vertex i,  cnnij is the number of common neighbors between vertices i and j, and aij is the element of the adjacency matrix, reflecting the number of edges across vertices i and j.

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Fig. 7

Starting from the subgraphs constructed on the first vertex group, one group is added at each step. For the new group, the unipartite graphs are constructed on the vertices of the group. As well, bipartite subgraphs are constructed between the vertices of the group and the vertices of groups preceding it, one at a time.

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Fig. 8

The divide-and-conquer approach for specifying the elements of the rectangular submatrix

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Fig. 9

An example of a graph for a link family

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Fig. 10

Distributing five binary vertices among the edges of a contracted topology

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Fig. 11

An example of a linkage arrangement for the Watt topology with links selected from a given link soup

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Fig. 12

Finding a spanning tree for the Watt topology of the six-bar linkages. Every (topological) edge that is not present in the spanning tree (shown with dotted lines) closes a loop, and is called a chord.

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Fig. 13

Determining the reference geometry for the linkage arrangement for the Watt topology of the six-bar linkage

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Fig. 14

(a) For revolute joint alignment, A1 must coincide with B1 and A2 must coincide with B2. (b) For prismatic joint alignment, α, β, d1, and d2 must be zero. (c) For clash to be avoided between a pair of spheres, d should be greater than R1+R2.

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Fig. 15

r-regular graphs with r > 2, with up to eight vertices, and containing only single edges

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Fig. 16

Average CPU time (measured in CPU CLOCKS) of the traditional and the proposed heuristic technique for isomorphism detection in 20 pairs of topologies

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Fig. 17

Closing open loops with known solutions

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Fig. 18

A seven-bar linkage is synthesized from rigid fragments of amino acids. Even though the linkage closes geometrically, the stability of the molecule is yet to be investigated.

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