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Research Papers: Design of Mechanisms and Robotic Systems

An Innovative Approach to Detect Isomorphism in Planar and Geared Kinematic Chains Using Graph Theory

[+] Author and Article Information
Vinjamuri Venkata Kamesh

Associate Professor
Mechanical Engineering,
Aditya Engineering College,
East Godavari District,
Surampalem 533 437, Andhra Pradesh, India
e-mail: kameshvv@gmail.com

Kuchibhotla Mallikarjuna Rao

Professor
Mechanical Engineering,
University College of Engineering,
JNTUK,
East Godavari District,
Kakinada 533 003, Andhra Pradesh, India
e-mail: rangaraokuchibhotla@gmail.com

Annambhotla Balaji Srinivasa Rao

Professor
Mechanical Engineering,
Sri Vasavi Institute of Engineering
and Technology,
Pedana Mandal,
Krishna District,
Nandamuru 521369, Andhra Pradesh India
e-mail: absrao71@gmail.com

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received February 7, 2017; final manuscript received August 3, 2017; published online October 3, 2017. Assoc. Editor: Massimo Callegari.

J. Mech. Des 139(12), 122301 (Oct 03, 2017) (11 pages) Paper No: MD-17-1109; doi: 10.1115/1.4037628 History: Received February 07, 2017; Revised August 03, 2017

Detection of isomorphism in planar and geared kinematic chains (GKCs) is an interesting area since many years. Enumeration of planar and geared kinematic chains becomes easy only when isomorphism problem is resolved effectively. Many researchers proposed algorithms based on topological characteristics or some coding which need lot of computations and comparisons. In this paper, a novel and simple algorithm is proposed based on graph theory by which elimination of isomorphic chains can be done very easily without any tedious calculations or comparisons. A new concept “Net distance” is proposed based on the graph theory to be a quantitative measure to assess isomorphism in planar kinematic chains (PKCs) as well as GKCs. The proposed algorithm is applied on nine-link two-degrees-of-freedom (DOF) distinct kinematic chains completely and the results are presented. Algorithm is tested on examples from eight-link 1-DOF, ten-link 1-DOF, 12-link 1-DOF, and 15link 4-DOF PKCs. The algorithm is also tested on four-, six-link 1-DOF GKCs to detect isomorphism. All the results are in agreement with the existing literature.

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Figures

Grahic Jump Location
Fig. 1

(a) Eight-link kinematic chain-1 and (b) eight-link kinematic chain-2

Grahic Jump Location
Fig. 2

(a) Graph for kinematic chain in Fig. 1(a) and (b) graph for kinematic chain in Fig. 1(b)

Grahic Jump Location
Fig. 3

(a) Ten-link kinematic chain-1 and (b) ten-link kinematic chain-2

Grahic Jump Location
Fig. 4

(a) Graph for kinematic chain in Fig. 3(a) and (b) graph for kinematic chain in Fig. 3(b)

Grahic Jump Location
Fig. 5

(a) 12-link kinematic chain-1, (b) 12-link kinematic chain-2, and (c) 12-link kinematic chain-3

Grahic Jump Location
Fig. 6

(a) Graph for kinematic chain in Fig. 5(a), (b) graph for kinematic chain in Fig. 5(b), and (c) graph for kinematic chain in Fig. 5(c)

Grahic Jump Location
Fig. 7

(a) 15-link kinematic chain-1, (b) 15-link kinematic chain-2, and (c) 15-link kinematic chain-3

Grahic Jump Location
Fig. 8

(a) Graph for kinematic chain in Fig. 7(a), (b) graph for kinematic chain in Fig. 7(b), and (c) graph for kinematic chain in Fig. 7(c)

Grahic Jump Location
Fig. 9

(a) Six-link EGT-1 and (b) six-link EGT-2

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