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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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References

Figures

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Fig. 1

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

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Fig. 2

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

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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. 5

Coordinate system configurations for rule–rule 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. 7

Coordinate systems configuration for rule–axis case

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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. 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. 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. 13

Variable helicoid–helicoid intersection 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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