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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
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
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

## Abstract

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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## References

Bennett, G. T. , 1903, “A New Mechanism,” Engineering, 76, pp. 777–778.
Bennett, G. T. , 1905, “The Parallel Motion of Sarrut and Some Allied Mechanisms,” Philos. Mag., 6(54), pp. 803–810.
Dai, J. S. , Huang, Z. , and Lipkin, H. , 2006, “Mobility of Overconstrained Parallel Mechanisms,” ASME J. Mech. Des., 128(1), pp. 220–229.
Dai, J. S. , 2014, Geometrical Foundations and Screw Algebra for Mechanisms and Robotics, Higher Education Press, Beijing, China (translated from Dai, J. S. 2019, Screw Algebra and Kinematic Approaches for Mechanisms and Robotics, Springer, London).
Kutzbach, K. , 1929, “Mechanische Leitungsverzweigung, Ihre Gesetze Und Anwendungen,” Maschinenbau, 8(21), pp. 710–716.
Myard, F. E. , 1931, “Contribution à La Géométrie Des Systèmes Articulés,” Soc. Math. France, 59, pp. 183–210.
Goldberg, M. , 1943, “New Five-Bar and Six-Bar Linkages in Three Dimensions,” ASME Trans., 65, pp. 649–661.
Sarrus, P. T. , 1853, “Note Sur La Transformation Des Mouvements Rectilignes Alternatifs, En Mouvements Circulaires, Et Reciproquement,” Acad. Sci., 36, pp. 1036–1038.
Bricard, R. , 1897, “Mémoire Sur La Théorie De L'octaedre Articulé,” J. Pure Appl. Math., 3, pp. 113–148.
Bricard, R. , 1927, Leçons De Cinématique, Gauthier-Villars, Paris, France.
Schatz, P. , 1942, “Mechanism Producing Wavering and Rotating Movements of Receptacles,” U.S. Patent No. US2302804.
Lee, C. C., and Dai, J. S., 2003, “Configuration Analysis of the Schatz Linkage,” J. Mech. Eng. Sci., 217(7), pp. 779–786.
Grodzinski, P., and M'Ewen, E., 1954, “Link Mechanisms in Modern Kinematics,” Proc. Inst. Mech. Eng., 168(1), pp. 877–896.
Waldron, K. J. , 1968, “Hybrid Overconstrained Linkages,” J. Mech, 3(2), pp. 73–78.
Wohlhart, K. , 1991, “Merging Two General Goldberg 5R Linkages to Obtain a New 6R Space Mechanism,” Mech. Mach. Theory, 26(7), pp. 659–668.
Chen, Y. , and You, Z. , 2007, “Spatial 6R Linkages Based on the Combination of Two Goldberg 5R Linkages,” Mech. Mach. Theory, 42(11), pp. 1484–1498.
Song, C. Y. , Chen, Y. , and Chen, I. M. , 2013, “A 6R Linkage Reconfigurable Between the Line-Symmetric Bricard Linkage and the Bennett Linkage,” Mech. Mach. Theory, 70, pp. 278–292.
Zhang, K. , and Dai, J. S. , 2014, “Trifurcation of the Evolved Sarrus-Motion Linkage Based on Parametric Constraints,” Advances in Robot Kinematics, Springer International Publishing, Cham, Switzerland, pp. 345–353.
Baker, J. E. , 1979, “The Bennett, Goldberg and Myard Linkages—in Perspective,” Mech. Mach. Theory, 14(4), pp. 239–253.
Baker, J. E. , 1980, “An Analysis of the Bricard Linkages,” Mech. Mach. Theory, 15(4), pp. 267–286.
Baker, J. E. , 2002, “Displacement–Closure Equations of the Unspecialised Double-Hooke's-Joint Linkage,” Mech. Mach. Theory, 37(10), pp. 1127–1144.
Cui, L. , and Dai, J. S. , 2011, “Axis Constraint Analysis and Its Resultant 6R Double-Centered Over-Constrainted Mechanisms,” ASME J. Mech. Rob., 3(3), p. 031004.
Kong, X. , 2014, “Type Synthesis of Single-Loop Over-Constrainted 6R Spatial Mechanisms for Circular Translation,” ASME J. Mech. Rob., 6(4), p. 041016.
Zhang, K. , Müller, A. , and Dai, J. S. , 2016, “A Novel Reconfigurable 7R Linkage With Multifurcation,” Advances in Reconfigurable Mechanisms and Robots II, Springer International Publishing, Cham, Switzerland, pp. 15–25.
Aimedee, F. , Gogu, G. , Dai, J. S. , Bouzgarrou, C. , and Bouton, N. , 2016, “Systematization of Morphing in Reconfigurable Mechanisms,” Mech. Mach. Theory, 96, pp. 215–224.
Zhang, K. , and Dai, J. S. , 2014, “A Kirigami-Inspired 8R Linkage and Its Evolved Overconstrained 6R Linkages With the Rotational Symmetry of Order Two,” ASME J. Mech. Rob., 6(2), p. 021007.
Barton, L. O. , 1993, Mechanism Analysis: Simplified Graphical and Analytical Techniques, Marcel Dekker, New York.
Waldron, K. J. , and Kinzel, G. L. , 1998, Kinematics, Dynamics, and Design of Machinery, Wiley, Hoboken, NJ.
Qin, Y. , Dai, J. S. , and Gogu, G. , 2014, “ Multi-Furcation in a Derivative Queer-Square Mechanism,” Mech. Mach. Theory, 81, pp. 36–53.
Kong, X. , 2015, “Kinematic Analysis of a 6R Single-Loop Overconstrained Spatial Mechanism for Circular Translation,” Mech. Mach. Theory, 93, pp. 163–174.
Zlatanov, D. , 1999, “Generalized Singularity Analysis of Mechanisms,” Ph.D. dissertation, University of Toronto, Toronto, Canada.
Zlatanov, D. , Bonev, I. A. , and Gosselin, C. M. , 2002, “Constraint Singularities as C-Space Singularities,” Advances in Robot Kinematics, Springer, Dordrecht, The Netherlands, pp. 183–192.
Harold Hilton, M. A. , 1920, Plane Algebraic Curves, The Clarendon Press, Oxford, UK.
Dhami, H. S. , 2009, Differential Calculus, New Age International Pvt Ltd Publishers, New Delhi, India.
Zhang, K. , and Dai, J. S. , 2015, “Screw-System-Variation Enabled Reconfiguration of the Bennett Plano-Spherical Hybrid Linkage and Its Evolved Parallel Mechanism,” ASME J. Mech. Des., 137(6), p. 062303.
Dai, J. S. , 2012, “Finite Displacement Screw Operators With Embedded Chasles' Motion,” ASME J. Mech. Rob., 4(4), p. 041002.
Dai, J. S. , 2014, Screw Algebra and Lie Groups and Lie Algebra, Higher Education Press, Beijing, China.
McCarthy, J. M. , 1990, An Introduction to Theoretical Kinematics, MIT Press, Cambridge, MA.
Baker, J. E. , 2003, “Overconstrained Six-Bars With Parallel Adjacent Joint-Axes,” Mech. Mach. Theory, 38(2), pp. 103–117.

## Figures

Fig. 1

Fig. 2

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

Fig. 3

Mapping from a free joint space to a configuration torus

Fig. 4

Curves and four bifurcation points on configuration torus

Fig. 5

General configuration of motion branch B

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)

Fig. 7

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

Fig. 8

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

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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