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Research Papers

A Novel Paradigm for the Qualitative Synthesis of Simple Kinematic Chains Based on Complexity Measures

[+] Author and Article Information
Waseem A. Khan1

Mem. ASME Senior Researcher  Jabez Technologies Inc., 5929 Rte Transcanadienne, suite 320, St. Laurent QC, H4T 1Z6 Canada e-mail: waseem@robotmaster.com

Jorge Angeles

Fellow ASME Professor Department of Mechanical Engineering,  McGill University, 817, Sherbrooke Street West, Montreal, QC H3A 2K6 Canada e-mail: angeles@cim.mcgill.ca

A point on a planar curve is called singular if two or more branches of the same curve meet. If the tangents at the meeting point are distinct, the point is called a node; if the tangents coincide, the point is called a cusp [14].

We denote with ‖ · ‖2 the 2-norm in a space of functions [16].

The proof follows directly from Eq. 9 if we replace k by km. Further, since m is constant for a given curve, it may be taken out of the summation, which proves the Lemma.

The reader is referred to any elementary differential geometry text for details on the shape operator

We acknowledge Prof. Grigore Gogu of Institut français de mécanique avancée, Clemont-Ferrand, France, for bringing this work to our attention.

G2 -continuity of a surface means continuity of the surface itself, of its normal, and of its curvatures.

P pairs not having an axis, only a direction is included here, with the provision that the discussion, in this case, pertains to their motion-direction.

Ternary and higher-order links can be accommodated, but the discussion of these links is left aside because of the scope of the paper.

1

Corresponding author.

J. Mechanisms Robotics 3(3), 031010 (Aug 12, 2011) (11 pages) doi:10.1115/1.4004226 History: Received February 19, 2010; Revised May 02, 2011; Published August 12, 2011; Online August 12, 2011

Proposed in this paper is a paradigm for the qualitative synthesis of simple kinematic chains that is based on the concept of complexity. Qualitative synthesis is understood here as the number and the type stages of the kinematic-synthesis process. The formulation hinges on the geometric complexity of the surface associated with lower kinematic pairs. First, the geometric complexity of curves and surfaces is recalled, as defined via the loss of regularity (LOR). The LOR, based in turn on the concept of diversity, measures the spectral richness of the curvature of either the curve or the surface under study. The paper closes with a complexity analysis of all six lower kinematic pairs, as a means to guide the mechanical designer into the conceptual stage of the design process. The paradigm is illustrated with the computation of the complexity of the four-bar linkage in all its versions, planar, spherical, and spatial, as well as that of a transmission for the conversion of a rotation about a vertical axis into one about a horizontal axis.

FIGURES IN THIS ARTICLE
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Copyright © 2011 by by ASME
Topics: Linkages , Chain
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References

Figures

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Figure 1

(a) A contour with two singular points and (b) its curvature distribution

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Figure 2

An extruded surface

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Figure 3

A surface of revolution

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Figure 4

The simplest surface of revolution common to the two links coupled by a R joint: (a) a 3D rendering of the surface SR and (b) its LOR versus. shaft radius r

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Figure 5

Cross section of the prismatic pair

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Figure 6

3D rendering of a screw based on a 2–4–6 polynomial: (a) p < λ and (b) p > λ

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Figure 7

(a) 3D plot of LOR versus p/λ and r/λ and (b) 2D plot of LOR versus r/λ for p/λ = 1.0

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Figure 8

Binary tree displaying possible link morphologies

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Figure 9

The general layout of a pan-tilt generator, showing its two input angular velocities, ωR and ωS , and its pan and tilt output angular velocities

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Figure 10

A schematic of the double universal joint converting rotation about a vertical into rotation about a horizontal axis

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Figure 11

A schematic of the RHRRR five-bar linkage.

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