Research Papers

Branch Reconfiguration of Bricard Linkages Based on Toroids Intersections: Line-Symmetric Case

[+] Author and Article Information
P. C. López-Custodio

Centre for Robotics Research,
King's College London,
University of London,
London WC2R 2 LS, UK
e-mail: pablo.lopez-custodio@kcl.ac.uk

J. S. Dai

Centre for Robotics Research,
King's College London,
University of London,
London WC2R 2 LS, UK
e-mail: jian.dai@kcl.ac.uk

J. M. Rico

Mechanical Engineering Department,
DICIS, Universidad de Guanajuato,
Salamanca 36500, Guanajuato, Mexico
e-mail: jrico@ugto.mx

1Corresponding author.

Manuscript received June 22, 2017; final manuscript received December 14, 2017; published online March 23, 2018. Assoc. Editor: Raffaele Di Gregorio.

J. Mechanisms Robotics 10(3), 031003 (Mar 23, 2018) (11 pages) Paper No: JMR-17-1188; doi: 10.1115/1.4038981 History: Received June 22, 2017; Revised December 14, 2017

This paper for the first time investigates a family of line-symmetric Bricard linkages by means of two generated toroids and reveals their intersection that leads to a set of special Bricard linkages with various branches of reconfiguration. The discovery is made in the concentric toroid–toroid intersection. By manipulating the construction parameters of the toroids, all possible bifurcation points are explored. This leads to the common bi-tangent planes that present singularities in the intersection set. The study reveals the presence of Villarceau and secondary circles in the toroid–toroid intersection. Therefore, a way to reconfigure the Bricard linkage to a pair of different types of Bennett linkage is uncovered. Further, a linkage with two Bricard and two Bennett motion branches is explored. In addition, the paper reveals the Altmann linkage as a member of the family of special line-symmetric Bricard linkage studied in this paper. The method is applied to the plane-symmetric case in the following paper published together with this paper.

Copyright © 2018 by ASME
Topics: Linkages
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Bricard, R. , 1897, “Mémoire sur la thèorie de l'octaèdre articulè,” J. Pure Appl. Math., 3, pp. 113–148.
Bricard, R. , 1927, Leçons de cinématique, Gauthier-Villars, Paris, France.
Waldron, K. J. , 1969, “Symmetric Overconstrained Linkages,” ASME J. Eng. Ind., 91(1), pp. 158–162. [CrossRef]
Hunt, K. H. , 1967, “Screw Axes and Mobility in Spatial Mechanisms Via the Linear Complex,” J. Mech., 2(3), pp. 307–327. [CrossRef]
Phillips, J. , 1990, Freedom in Machinery (Screw Theory Exemplified, Vol. 2), Cambridge University Press, Cambridge, UK.
Baker, J. E. , 1980, “An Analysis of the Bricard Linkages,” Mech. Mach. Theory, 15(4), pp. 267–286. [CrossRef]
Mavroidis, C. , and Roth, B. , 1995, “Analysis of Overconstrained Mechanisms,” ASME J. Mech. Des., 117(1), pp. 69–74. [CrossRef]
Mavroidis, C. , and Roth, B. , 1995, “New and Revised Overconstrained Mechanisms,” ASME J. Mech. Des., 117(1), pp. 75–82. [CrossRef]
Dai, J. , Huang, Z. , and Lipkin, H. , 2006, “Mobility of Overconstrained Parallel Mechanisms,” ASME J. Mech. Des., 128(1), pp. 220–229. [CrossRef]
Dai, J. S. , Huang, Z. , and Lipkin, H. , 2004, “Screw System Analysis of Parallel Mechanisms and Applications to Constraint and Mobility Study,” ASME Paper No. DETC2004-57604.
Phillips, J. , 1984, Freedom in Machinery (Introducing Screw Theory, Vol. 1), Cambridge University Press, Cambridge, UK.
Hunt, K. H. , 1978, Kinematic Geometry of Mechanisms, Oxford University Press, New York.
Hunt, K. H. , 1968, “Note on Complexes and Mobility,” J. Mech., 3(3), pp. 199–202. [CrossRef]
Baker, J. E. , and Wohlhart, K. , 1994, “On the Single Screw Reciprocal to the General Line-Symmetric Six-Screw Linkage,” Mech. Mach. Theory, 29(1), pp. 169–175. [CrossRef]
Jenkins, E. M. , Crossley, F. R. E. , and Hunt, K. H. , 1969, “Gross Motion Attributes of Certain Spatial Mechanisms,” ASME J. Eng. Ind., 91(1), pp. 83–90. [CrossRef]
Torfason, L. E. , and Crossley, F. R. E. , 1971, “Use of the Intersection of Surfaces as a Method for Design of Spatial Mechanisms,” Third World Congress for the Theory of Machines and Mechanisms, Kupari, Yugoslavia, Sept. 13–20, Paper No. B-20, pp. 247–258.
Torfason, L. E. , and Sharma, A. K. , 1973, “Analysis of Spatial RRGRR Mechanisms by the Method of Generated Surfaces,” ASME J. Eng. Ind., 95(3), pp. 704–708. [CrossRef]
Shrivastava, A. K. , and Hunt, K. H. , 1973, “Dwell Motion From Spatial Linkages,” ASME J. Eng. Ind., 95(2), pp. 511–518. [CrossRef]
Liu, Y. , and Zsombor-Murray, P. , 1995, “Intersection Curves Between Quadric Surfaces of Revolution,” Trans. Can. Soc. Mech. Eng., 19(4), pp. 435–453.
Hunt, K. H. , 1973, “Constant-Velocity Shaft Couplings: A General Theory,” ASME J. Eng. Ind., 95(2), pp. 455–464. [CrossRef]
Lee, C. C. , and Hervé, J. M. , 2012, “A Discontinuously Movable Constant Velocity Shaft Coupling of Koenigs Joint Type,” Advances in Reconfigurable Mechanisms and Robots I, M. Z. J. S. Dai and X. Kong, eds., pp. 35–43.
Su, H. J. , and McCarthy, J. M. , 2005, “Dimensioning a Constrained Parallel Robot to Reach a Set of Task Positions,” IEEE International Conference on Robotics and Automation (ICRA), Barcelona, Spain, Apr. 18–22, pp. 4026–4030.
Fichter, E. F. , and Hunt, K. H. , 1975, “The Fecund Torus, Its Bitangent-Circles and Derived Linkages,” Mech. Mach. Theory, 10(2–3), pp. 167–176. [CrossRef]
López-Custodio, P. C. , Rico, J. M. , Cervantes-Sánchez, J. J. , and Pérez-Soto, G. , 2016, “Reconfigurable Mechanisms From the Intersection of Surfaces,” ASME J. Mech. Rob., 8(2), p. 021029. [CrossRef]
López-Custodio, P. C. , Rico, J. M. , and Cervantes-Sánchez, J. J. , 2017, “Local Analysis of Helicoid-Helicoid Intersections in Reconfigurable Linkages,” ASME J. Mech. Rob., 9(3), p. 031008. [CrossRef]
Dai, J. S. , and Gogu, G. , 2016, “Special Issue on Reconfigurable Mechanisms: Morphing, Metamorphosis and Reconfiguration Through Constraint Variations and Reconfigurable Joints,” Mech. Mach. Theory, 96(Pt. 2), pp. 213–214. [CrossRef]
Kuo, C. H. , Dai, J. S. , and Yan, H. S. , 2009, “Reconfiguration Principles and Strategies for Reconfigurable Mechanisms,” ASME/IFToMM International Conference on Reconfigurable Mechanisms and Robots (ReMAR 2009), London, June 22–24, pp. 1–7. http://ieeexplore.ieee.org/document/5173802/
Wohlhart, K. , 1996, “Kinematotropic Linkages,” Recent Advances in Robot Kinematics, J. Lenarčič and V. Parenti-Castelli , eds., Dordrecht, The Netherlands, pp. 359–368. [CrossRef]
Galletti, C. , and Fanghella, P. , 2001, “Single-Loop Kinematotropic Mechanisms,” Mech. Mach. Theory, 36(3), pp. 743–761. [CrossRef]
Dai, J. S. , and Jones, J. R. , 1999, “Mobility in Metamorphic Mechanisms of Foldable/Erectable Kinds,” ASME J. Mech. Des., 121(3), pp. 375–382. [CrossRef]
Zhang, K. , Dai, J. S. , and Fang, Y. , 2012, “Geometric Constraint and Mobility Variation of Two 3 SvPSv Metamorphic Parallel Mechanisms,” ASME J. Mech. Des., 135(1), p. 011001. [CrossRef]
Gan, D. , Dai, J. S. , Dias, J. , and Lakmal, S. , 2013, “Unified Kinematics and Singularity Analysis of a Metamorphic Parallel Mechanism With Bifurcated Motion,” ASME J. Mech. Rob., 5(3), p. 031004. [CrossRef]
Li, S. , and Dai, J. S. , 2012, “Structure Synthesis of Single-Driven Metamorphic Mechanisms Based on the Augmented Assur Groups,” ASME J. Mech. Rob., 4(3), p. 031004. [CrossRef]
Kong, X. , 2014, “Reconfiguration Analysis of a 3-DOF Parallel Mechanism Using Euler Parameter Quaternions and Algebraic Geometry Method,” Mech. Mach. Theory, 74, pp. 188–201. [CrossRef]
Kong, X. , 2012, “Type Synthesis of Variable Degrees-of-Freedom Parallel Manipulators With Both Planar and 3T1R Operation Modes,” ASME Paper No. DETC2012-70621.
Kong, X. , and Pfurner, M. , 2015, “Type Synthesis and Reconfiguration Analysis of a Class of Variable-DOF Single-Loop Mechanisms,” Mech. Mach. Theory, 85, pp. 116–128. [CrossRef]
Ye, W. , Fang, Y. , Zhang, K. , and Guo, S. , 2014, “A New Family of Reconfigurable Parallel Mechanisms With Diamond Kinematotropic Chain,” Mech. Mach. Theory, 74, pp. 1–9. [CrossRef]
Zhang, K. , and Dai, J. S. , 2016, “Geometric Constraints and Motion Branch Variations for Reconfiguration of Single-Loop Linkages With Mobility One,” Mech. Mach. Theory, 106, pp. 16–29. [CrossRef]
Zhang, K. , Müller, A. , and Dai, J. S. , 2016, “A Novel Reconfigurable 7R Linkage With Multifurcation,” Advances in Reconfigurable Mechanisms and Robots II, X. Ding , X. Kong , and J. S. Dai , eds., Springer International Publishing, Cham, Switzerland, pp. 3–14. [CrossRef]
Qin, Y. , Dai, J. , and Gogu, G. , 2014, “Multi-Furcation in a Derivative Queer-Square Mechanism,” Mech. Mach. Theory, 81(11), pp. 36–53. [CrossRef]
López-Custodio, P. C. , Rico, J. M. , Cervantes-Sánchez, J. J. , Pérez-Soto, G. I. , and Díez-Martínez, C. R. , 2017, “Verification of the Higher Order Kinematic Analyses Equations,” Eur. J. Mech. A, 61, pp. 198–215. [CrossRef]
Müller, A. , 2016, “Local Kinematic Analysis of Closed-Loop Linkages Mobility, Singularities, and Shakiness,” ASME J. Mech. Rob., 8(4), p. 041013. [CrossRef]
Müller, A. , 2005, “Geometric Characterization of the Configuration Space of Rigid Body Mechanisms in Regular and Singular Points,” ASME Paper No. DETC2005-84712.
Aimedee, F. , Gogu, G. , Dai, J. , Bouzgarrou, C. , and Bouton, N. , 2016, “Systematization of Morphing in Reconfigurable Mechanisms,” Mech. Mach. Theory, 96(Pt. 2), pp. 215–224. [CrossRef]
Kong, X. , 2017, “Reconfiguration Analysis of Multimode Single-Loop Spatial Mechanisms Using Dual Quaternions,” ASME J. Mech. Rob., 9(5), p. 031004. [CrossRef]
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. [CrossRef]
Chen, Y. , and Chai, W. H. , 2011, “Bifurcation of a Special Line and Plane Symmetric Bricard Linkage,” Mech. Mach. Theory, 46(4), pp. 515–533. [CrossRef]
Zhang, K. , and Dai, J. , 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. [CrossRef]
Lu, S. , Zlatanov, D. , Ding, X. , Zoppi, M. , and Guest, S. D. , 2016, “Reconfigurable Chains of Bifurcating Type III Bricard Linkages,” Advances in Reconfigurable Mechanisms and Robots II, X. Ding , X. Kong , and J. S. Dai , eds., Springer International Publishing, Cham, Switzerland, pp. 3–14. [CrossRef]
Qi, X. , Huang, H. , Miao, Z. , Li, B. , and Deng, Z. , 2016, “Design and Mobility Analysis of Large Deployable Mechanisms Based on Plane-Symmetric Bricard Linkage,” ASME J. Mech. Rob., 139(2), p. 022302. [CrossRef]
Chen, Y. , You, Z. , and Tarnai, T. , 2005, “Threefold-Symmetric Bricard Linkages for Deployable Structures,” Int. J. Solids Struct., 42(8), pp. 2287–2301. [CrossRef]
Myard, F. E. , 1931, “Contribution á la géométrie des systèmes articulés,” Soc. Math. France, 59, pp. 183–210. [CrossRef]
Bricard, R. , 1925, “Démonstration élémentaires de propriétés fondamentales du tore,” Nouv. Ann. Math., 3, pp. 308–313.
Lee, C.-C. , and Hervé, J. M. , 2014, “Oblique Circular Torus, Villarceau Circles, and Four Types of Bennett Linkages,” Proc. Inst. Mech. Eng., Part C, 228(4), pp. 742–752. [CrossRef]
Bil, T. , 2012, “Analysis of the Bennett Linkage in the Geometry of Tori,” Mech. Mach. Theory, 53, pp. 122–127. [CrossRef]
Baker, J. , 1984, “On 5-Revolute Linkages With Parallel Adjacent Joint Axes,” Mech. Mach. Theory, 19(6), pp. 467–475. [CrossRef]
Bil, T. , and Budniak, Z. , 2014, “Model of 5R Spatial Linkages in Geometry of Tori,” Int. J. Appl. Mech. Eng., 19(4), pp. 823–830. [CrossRef]
Bil, T. , 2011, “Kinematic Analysis of a Universal Spatial Mechanism Containing a Higher Pair Based on Tori,” Mech. Mach. Theory, 46(4), pp. 412–424. [CrossRef]
Chung, W. Y. , 2005, “Mobility Analysis of RSSR Mechanisms by Working Volume,” ASME J. Mech. Des., 127(1), pp. 156–159. [CrossRef]
López-Custodio, P. C. , Dai, J. S. , and Rico, J. M. , 2018, “Branch Reconfiguration of Bricard Linkages Based on Toroids Intersections: Plane-Symmetric Case,” ASME J. Mech. Rob., 10(3), p. 031002.
Müller, A. , 2009, “Generic Mobility of Rigid Body Mechanisms,” Mech. Mach. Theory, 44(6), pp. 1240–1255. [CrossRef]
Müller, A. , 2015, “Representation of the Kinematic Topology of Mechanisms for Kinematic Analysis,” Mech. Mach. Theory, 6, pp. 137–146.
Cox, D. A. , Little, J. B. , and O'Shea, D. , 2007, Ideals, Varieties and Algorithms, Springer, New York. [CrossRef]
Arponen, T. , Müller, A. , Piipponen, S. , and Tuomela, J. , 2014, “Kinematical Analysis of Overconstrained and Underconstrained Mechanisms by Means of Computational Algebraic Geometry,” Meccanica, 49(4), pp. 843–862. [CrossRef]
Liu, X.-M. , Liu, C.-Y. , Yong, J.-H. , and Paul, J.-C. , 2011, “Torus/Torus Intersection,” Comput.-Aided Des. Appl., 8(3), pp. 465–477. [CrossRef]
Song, C.-Y. , Chen, Y. , and Chen, I.-M. , 2014, “Kinematic Study of the Original and Revised General Line-Symmetric Bricard 6R Linkages,” ASME J. Mech. Rob., 6(3), p. 031002. [CrossRef]
Villarceau, Y. , 1848, “Théorème sur le tore,” Nouv. Ann. Math., 7, pp. 345–347.
Bennett, G. T. , 1903, “A New Mechanism,” Engineering, 76, pp. 777–778.
Lee, C. C. , and Hervé, J. M. , 2015, “The Metamorphic Bennett Linkages,” 14th IFToMM World Congress, Taipei, Taiwan, Oct. 25--30, pp. 394–399.
Altmann, J. G. , 1954, “Communications to Grodzinski P. and Mewen E.: Link Mechanisms in Modern Kinematics,” Proc. Inst. Mech. Eng., 168(1), pp. 877–896. [CrossRef]
Baker, J. E. , 1993, “A Geometrico-Algebraic Exploration of Altmann's Linkage,” Mech. Mach. Theory, 28(2), pp. 249–260. [CrossRef]
Baker, J. E. , 2012, “On the Closure Modes of a Generalised Altmann Linkage,” Mech. Mach. Theory, 52, pp. 243–247. [CrossRef]
Cui, L. , and Dai, J. , 2011, “Axis Constraint Analysis and Its Resultant 6R Double-Centered Overconstrained Mechanisms,” ASME J. Mech. Rob., 3(3), p. 031004. [CrossRef]


Grahic Jump Location
Fig. 3

The line-symmetric Bricard linkages that can be analyzed as the intersection of two concentric toroids

Grahic Jump Location
Fig. 2

An RR dyad generating a general toroid

Grahic Jump Location
Fig. 1

The general line-symmetric case of Bricard linkage

Grahic Jump Location
Fig. 4

Diagram for the line-symmetric case with sA=sB=0. (a) Assembly with vA=−vB and (b) assembly with vA = vB.

Grahic Jump Location
Fig. 8

An example of reconfigurable Bricard linkage, which can evolve to a pair different types of Bennett linkage. The intersection of the two concentric toroids is composed of four circles Ci, i=1,…,4. The bifurcation points are labeled as Pi, i=1,…,6. The linkage is shown in a singular configuration with E = P5.

Grahic Jump Location
Fig. 9

Different configurations for an example of line-symmetric linkage where the generated toroids intersect in two circles

Grahic Jump Location
Fig. 6

Cases with circles (in bold curves) in the intersection. (a) Villarceau circles and (b) secondary circles. In both cases, points 1–4 are permanent points of tangency. In (b) 5 and 6 are points of tangency in the XZ plane.

Grahic Jump Location
Fig. 7

A toroid and a rotated copy of itself sharing secondary circles: (a) θ=2γ* and (b) θ=π−2γ*

Grahic Jump Location
Fig. 10

The four curves of uA versus uB obtained from the motion branches of the linkage. Bifurcation points are located in relation with Fig. 8. (a) Toroidal representation and (b) Cartesian plot.

Grahic Jump Location
Fig. 5

Intersection of toroids A and B and their bi-tangent plane with the XZ plane




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