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Research Papers

Motion Cycle and Configuration Torus With Their Relationship to Furcation During Reconfiguration

[+] Author and Article Information
Xuesi Ma

MoE Key Laboratory for Mechanism Theory and
Equipment Design,
International Centre for
Advanced Mechanisms and Robotics,
School of Mechanical Engineering
Tianjin University,
Tianjin 300072, China
e-mail: maxuesi@tju.edu.cn

Xinsheng Zhang

Center for Robotics Research,
King's College London,
Strand WC2R 2 LS, London, UK
e-mail: xinsheng.zhang@kcl.ac.uk

Jian S. Dai

Chair of Mechanisms and Robotics,
MoE Key Laboratory for Mechanism Theory
and Equipment Design,
International Centre for
Advanced Mechanisms and Robotics,
Tianjin University,
Tianjin 300072, China;
Center for Robotics Research,
King's College London,
Strand WC2R 2 LS, London, UK
e-mail: jian.dai@kcl.ac.uk

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received October 31, 2017; final manuscript received May 10, 2018; published online July 9, 2018. Assoc. Editor: Philippe Wenger.

J. Mechanisms Robotics 10(5), 051006 (Jul 09, 2018) (10 pages) Paper No: JMR-17-1374; doi: 10.1115/1.4040357 History: Received October 31, 2017; Revised May 10, 2018

In a classical mobility-one single loop linkage, the motion begins from an original position determined by the assembled condition and runs in cycles. In normal circumstances, the linkage experiences a full cycle when the input-joint completes a full revolution. However, there are some linkages that accomplish a whole cycle with the input-joint having to go through multiple revolutions. Their motion cycle covers multiple revolutions of the input-joint. This paper investigates this typical phenomenon that the output angle is in a different motion cycle of the input angle that we coin this as the multiple input-joint revolution cycle. The paper then presents the configuration torus for presenting the motion cycle and reveals both bifurcation and double points of the linkage, using these mathematics-termed curve characteristics for the first time in mechanism analysis. The paper examines the motion cycle of the Bennett plano-spherical hybrid linkage that covers an 8π range of an input-joint revolution, reveals its four double points in the kinematic curve, and presents two motion branches in the configuration torus where double points give bifurcations of the linkage. The paper further examines the Myard plane-symmetric 5R linkage with its motion cycle covering a 4π range of the input-joint revolution. The paper, hence, presents a way of mechanism cycle and reconfiguration analysis based on the configuration torus.

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Figures

Grahic Jump Location
Fig. 1

D–H parameters of the links

Grahic Jump Location
Fig. 2

Bennett plano-spherical hybrid 6R linkage and its parameters: (a) the model of the linkage and (b) the parameters of the linkage

Grahic Jump Location
Fig. 3

Mapping from a free joint space to a configuration torus

Grahic Jump Location
Fig. 4

Curves and four bifurcation points on configuration torus

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Fig. 5

General configuration of motion branch B

Grahic Jump Location
Fig. 6

Bifurcations of Bennett plano-spherical hybrid linkage at different positions: (a) configuration 1: equivalent spherical 4R linkage when θ1=−(π/2), (b) configuration 2: equivalent spherical 4R linkage when θ1=(3π/2), (c) configuration 3: equivalent spherical 4R linkage when θ1=(7π/2), and (d) configuration 4: equivalent spherical 4R linkage when θ1=(11π/2)

Grahic Jump Location
Fig. 7

Myard plane-symmetric 5R linkage and its parameters: (a) the model of the linkage and (b) the parameters of the linkage

Grahic Jump Location
Fig. 8

A motion cycle on configuration torus of Myard plane-symmetric 5R linkage

Grahic Jump Location
Fig. 9

Configurations of Myard plane-symmetric 5R linkage at different positions: (a) full singularity 1 when θ1=π and θ5=0, (b) configuration A when θ1=5.234 and θ5=(π/2), (c) double point 1 with partial singularity when θ1=2π and θ5=(2/3π), (d) full singularity 2 when θ1=3π and θ5=π, (e) double point 2 with partial singularity when θ1=4π and θ5=(4/3π), and (f) configuration B when θ1=13.613 and θ5=(3π/2)

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