Research Papers

Local Analysis of Helicoid–Helicoid Intersections in Reconfigurable Linkages

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

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

J. M. Rico

Mechanical Engineering Department,
DICIS, Universidad de Guanajuato,
Salamanca, Guanajuato 36885, Mexico
e-mail: jrico@ugto.mx

J. J. Cervantes-Sánchez

Mechanical Engineering Department,
DICIS, Universidad de Guanajuato,
Salamanca, Guanajuato 36885, Mexico
e-mail: jecer@ugto.mx

1Corresponding author.

Manuscript received August 24, 2016; final manuscript received December 13, 2016; published online March 22, 2017. Assoc. Editor: Raffaele Di Gregorio.

J. Mechanisms Robotics 9(3), 031008 (Mar 22, 2017) (17 pages) Paper No: JMR-16-1245; doi: 10.1115/1.4035682 History: Received August 24, 2016; Revised December 13, 2016

Kinematic chains are obtained from the helicoid–helicoid intersections applying the method of surfaces generated by kinematic dyads. Some local properties of the helicoids are used to obtain the bifurcation points in the configuration space of the obtained kinematic chains. It is proven that certain relationships between the two helicoids lead to a periodic behavior of these bifurcations, which suggest that, if the kinematic pairs (P and H) could move without a limit, the kinematic chain would theoretically feature an infinity of operation modes. Finally, a mechanism which is able to change the helicoid–helicoid intersection curve during its motion is proven to change its finite mobility in one of its operation modes.

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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.
Wohlhart, K. , 1996, “Kinematotropic Linkages,” Recent Advances in Robot Kinematics, J. Lenarčič and V. Parent-Castelli , eds., Springer, Dordrecht, The Netherlands, pp. 359–368.
Galletti, C. , and Fanghella, P. , 2001, “ Single-Loop Kinematotropic Mechanisms,” Mech. Mach. Theory, 36(3), pp. 743–761. [CrossRef]
Zeng, Q. , Ehmann, K. F. , and Cao, J. , 2016, “ Design of General Kinematotropic Mechanisms,” Rob. Comput. Integr. Manuf., 38, pp. 67–81. [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.
Zlatanov, D. , Bonev, I. , and Gosselin, C. , 2002, “ Constraint Singularities as Configuration Space Singularities,” Advances in Robot Kinematics, J. Lenarčič and F. Thomas , eds., Springer, Dordrecht, The Netherlands, pp. 183–192.
Dai, J. , and Jones, J. R. , 1999, “ Mobility in Metamorphic Mechanisms of Foldable/Erectable Kinds,” ASME J. Mech. Des., 121(3), pp. 375–382. [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.
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]
Kong, X. , and Jin, Y. , 2016, “ Type Synthesis of 3-DOF Multi-Mode Translational/Spherical Parallel Mechanisms With Lockable Joints,” Mech. Mach. Theory, 96(Pt. 2), pp. 323–333. [CrossRef]
López-Custodio, P. , Rico, J. , Cervantes-Sánchez, J. , and Pérez-Soto, G. , 2016, “ Reconfigurable Mechanisms From the Intersection of Surfaces,” ASME J. Mech. Rob., 8(2), p. 021029. [CrossRef]
Torfason, L. , and Crossley, F. , 1971, “ Use of the Intersection of Surfaces as a Method for Design of Spatial Mechanisms,” 3rd World Congress for the Theory of Machines and Mechanisms, Vol. B, Kupari, Yugoslavia, pp. 247–258.
Torfason, L. , and Sharma, A. , 1973, “ Analysis of Spatial RRGRR Mechanisms by the Method of Generated Surfaces,” ASME J. Eng. Ind., 95(3), pp. 704–708. [CrossRef]
Shrivastava, A. , and Hunt, K. , 1973, “ Dwell Motion From Spatial Linkages,” ASME J. Eng. Ind., 95(2), pp. 511–518. [CrossRef]
Lee, C. , and Hervé, J. , 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., Springer-Verlag, London, pp. 35–43.
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]
Levin, J. , 1976, “ A Parametric Algorithm for Drawing Pictures of Solid Objects Composed of Quadric Surfaces,” Commun. ACM, 19(10), pp. 555–563. [CrossRef]
Levin, J. , 1979, “ Mathematical Models for Determining the Intersection of Quadric Surfaces,” Comput. Graphics Image Process., 11(1), pp. 73–87. [CrossRef]
Whitney, H. , 1965, “ Tangents to an Analytic Variety,” Ann. Math., 81(3), pp. 496–549. [CrossRef]
Müller, A. , 1998, “ 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.
Müller, A. , 2016, “ Local Kinematic Analysis of Closed-Loop Linkages Mobility, Singularities, and Shakiness,” ASME J. Mech. Rob., 8(4), p. 041013. [CrossRef]
Krivoshapko, S. , and Ivanov, V. , 2015, Encyclopedia of Analytical Surfaces, Springer, Cham, Switzerland.
Rico, J. , and Ravani, B. , 2003, “ On Mobility Analysis of Linkages Using Group Theory,” ASME J. Mech. Des., 125(1), pp. 70–80. [CrossRef]
Hervé, J. , 1978, “ Analyse Structurelle des Mécanismes par Groupe des Déplacements,” Mech. Mach. Theory, 13(4), pp. 437–450. [CrossRef]
Crane, C. , and Duffy, J. , 1998, Kinematic Analysis of Robot Manipulators, Cambridge University Press, Cambridge, UK.
Waldron, K. , 1967, “ A Family of Overconstrained Linkages,” J. Mech., 2(2), pp. 201–211. [CrossRef]
Lee, C. , and Hervé, J. , 2010, “ Mechanical Generators of 2-DoF Translation Along a Ruled Surface,” Advances in Robot Kinematics, J. Lenarčič and M. Stanisic , eds., Springer, Dordrecht, The Netherlands, pp. 73–80.
Su, H. , and McCarthy, J. , 2005, “ Dimensioning a Constrained Parallel Robot to Reach a Set of Task Positions,” IEEE International Conference on Robotics and Automation, Barcelona, Spain, Apr. 18–22, pp. 4026–4030.
Liu, Y. , and Zsombor-Murray, P. , 1995, “ Intersection Curves Between Quadric Surfaces of Revolution,” Trans. Can. Soc. Mech. Eng., 19(4), pp. 435–453.
Demazure, M. , 2000, Bifurcations and Catastrophes, Springer-Verlag, Berlin.
Lerbet, J. , 1998, “ Analytic Geometry and Singularities of Mechanisms,” Z. Angew. Math. Mech., 78(10), pp. 687–694. [CrossRef]
Müller, A. , 2002, “ Local Analysis of Singular Configuration of Open and Closed Loop Manipulators,” Multibody Syst. Dyn., 8(3), pp. 297–326. [CrossRef]
Müller, A. , 2014, “ Higher Derivatives of the Kinematic Mapping and Some Applications,” Mech. Mach. Theory, 76, pp. 70–85. [CrossRef]
Diez-Martínez, C. , Rico, J. , and Cervantes-Sánchez, J. , 2006, “ Mobility and Connectivity in Multiloop Linkages,” Advances in Robot Kinematics, J. Lenarčič and B. Roth , eds., Springer, Dordrecht, The Netherlands, pp. 455–464.
López-Custodio, P. , Rico, J. , Cervantes-Sánchez, J. , Pérez-Soto, G. , and Díez-Martínez, C. , 2017, “ Verification of the Higher Order Kinematic Analyses Equations,” Eur. J. Mech. - A/Solids, 61, pp. 198–215. [CrossRef]
Rico, J. , Gallardo, J. , and Duffy, J. , 1999, “ Screw Theory and the Higher Order Kinematic Analysis of Serial and Closed Chains,” Mech. Mach. Theory, 34(4), pp. 559–586. [CrossRef]
López-Custodio, P. , 2012, “ Análisis Cinemáticos de Orden Superior y Movilidad de Cadenas Cinemáticas,” B.Sc. thesis, Universidad de Guanajuato, Salamanca, Gto. Mexico.
Müller, A. , 2016, “ Recursive Higher-Order Constraints for Linkages With Lower Kinematic Pairs,” Mech. Mach. Theory, 100, pp. 33–43. [CrossRef]
Kreyszig, E. , 1959, Differential Geometry, University of Toronto Press, Toronto, ON, Canada.
Ye, X. , and Maekawa, T. , 1999, “ Differential Geometry of Intersection Curves of Two Surfaces,” Comput. Aided Geometric Des., 16(8), pp. 767–788. [CrossRef]
Barnhill, R. , and Kersey, S. , 1990, “ A Marching Method for Parametric Surface/Surface Intersection,” Comput. Aided Geometric Des., 7(1), pp. 257–280. [CrossRef]
Pérez-Soto, G. , and Tadeo, A. , 2006, “ Síntesis de Número de Cadenas Cinemáticas, un Nuevo Enfoque y Nuevas Herramientas Matemáticas,” M.Sc. thesis, Universidad de Guanajuato, Salamanca, Gto. Mexico.
Lee, C. , and Hervé, J. , 2016, “ Various Types of RC//-Like Linkages and the Discontinuously Movable Koenigs Joint,” Mech. Mach. Theory, 96(Pt. 2), pp. 255–268. [CrossRef]
Tadeo-Chávez, A. , Rico, J. , Cervantes-Sánchez, J. , Pérez-Soto, G. , and Müller, A. , 2011, “ Screw Systems Generated by Subalgebras: A Further Analysis,” ASME Paper No. DETC2011-48304.
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]


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Fig. 3

Helicoid–helicoid intersection mechanism: (a) HPSPH and (b) HPRRPH

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Fig. 4

Cases that locally may lead to confusion: (a) cylinder–cylinder, a single curve with self-crossing and (b) right helicoid-cone, an infinity of bifurcations that lead to the same two curves

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Fig. 2

Closed oblique helicoid: (a) parameterization and (b) kinematic chain generator

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Fig. 1

Open oblique helicoid: (a) parameterization and (b) kinematic chain generator

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Fig. 7

Coordinate systems configuration for rule–axis case

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Fig. 6

Mechanism obtained from the rule–rule case: (a) singular configuration, (b) regular configuration in V0, and (c) regular configuration in V1

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Fig. 9

Mechanism obtained from the axis–rule case: (a) general configuration in V0 and (b) and (c) regular configurations in two different operation modes

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Fig. 10

Special case with γA=γB=π/2, hB=−hA, and dA = 0

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Fig. 11

Coordinate systems configuration for axis–axis case

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Fig. 5

Coordinate system configurations for rule–rule case

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Fig. 13

Variable helicoid–helicoid intersection linkage

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Fig. 8

Surfaces for case 2 with γA=105π/180, γB=65π/180,hA=77/(2π), hA=110/(2π), and dA = 100: (a) Section {σA(uA,vA) | (uA,vA)∈[−π,π]×[−7000,7000]} intersecting {σB(uB,vB) |(uA,vA)∈[297.862−π,297.862+π]×[−7000,7000]}, the dashed curve does not intersect L0 since uB10−uB9 is barely bigger than π. (b) Same section of helicoid A intersecting ten periods of helicoid B, the part of C crossing L0 is shown in black curves. The intersection of equivalent cones at infinity is also shown in the figure. The values of uA tend to ±1.1797 and ±2.1232, while the values of uB tend to ±1.3992 + 2 and ±2.0048+2Nπ, ∀N∈Z.

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Fig. 12

Mechanism obtained from the axis–axis case. The black curve is the one in which X lies for the shown configuration of the linkage.

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Fig. 14

Trifurcation of the variable helicoid–helicoid intersection linkage at q1a1b(0) = 0

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Fig. 15

Special case of the variable helicoid–helicoid intersection mechanism with γ = h = 0

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Fig. 16

Trifurcation of the special case of the variable helicoid–helicoid intersection mechanism with γ = h = 0 for X = (0, d, 0) and q1 = 0. Equivalent diagrams can be drawn for {X = (0, −d, 0), q1 = 0}, {X = (0, d, 0), q1 = π}, and {X = (0, −d, 0), q1 = π}.



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