Box Approximations of Planar Linkage Configuration Spaces

[+] Author and Article Information
J. M. Porta

 Institut de Robòtica i Informàtica Industrial (UPC-CSIC), Llorens Artigas 4-6, 08028 Barcelonaporta@iri.upc.edu

L. Ros

 Institut de Robòtica i Informàtica Industrial (UPC-CSIC), Llorens Artigas 4-6, 08028 Barcelonaros@iri.upc.edu

T. Creemers

 Institut de Robòtica i Informàtica Industrial (UPC-CSIC), Llorens Artigas 4-6, 08028 Barcelonacreemers@iri.upc.edu

F. Thomas

 Institut de Robòtica i Informàtica Industrial (UPC-CSIC), Llorens Artigas 4-6, 08028 Barcelonathomas@iri.upc.edu

J. Mech. Des 129(4), 397-405 (Apr 18, 2006) (9 pages) doi:10.1115/1.2437808 History: Received February 03, 2006; Revised April 18, 2006

This paper presents a numerical method able to compute all possible configurations of planar linkages. The procedure is applicable to rigid linkages (i.e., those that can only adopt a finite number of configurations) and to mobile ones (i.e., those that exhibit a continuum of possible configurations). The method is based on the fact that this problem can be reduced to finding the roots of a polynomial system of linear, quadratic, and hyperbolic equations, which is here tackled with a new strategy exploiting its structure. The method is conceptually simple and easy to implement, yet it provides solutions of the desired accuracy in short computation times. Experiments are included that show its performance on the double butterfly linkage and on larger linkages formed by the concatenation of basic patterns.

Copyright © 2007 by American Society of Mechanical Engineers
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Figure 4

Output boxes at increasing resolution. The horizontal and vertical axes, respectively, correspond to cos(θ2) and cos(θ4), spanning the range [−1,1] in all cases, with marks separated 0.5 units apart

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Figure 1

Double butterfly linkage

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Figure 2

(a) Shrinking Bc to fit the linear variety L(v)=0, (b) half-planes approximating the circular arc inside Bc, and (c) smallest box enclosing the intersection of L(v)=0 with the half-planes in (b)

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Figure 3

The six real solutions of the double butterfly linkage for θ6=1.175rad(67.38deg), given in radians

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Figure 5

Path followed by point B of the double butterfly linkage (the ground link is rotated 90deg to the left with respect to Fig. 1). As observed, B may follow one of four different cyclic trajectories, T1, T2, T3, and T4, reflecting four (mobile) assembly modes for the mechanism. A sample configuration of the linkage following the fourth mode is also shown overlaid.

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Figure 6

The three-caterpillar framework (left), and the 3-RPR planar manipulator (right)

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Figure 7

The 64 solutions of the three-caterpillar

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Figure 8

Spiral-shaped construction of 200 repeated patterns

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Figure 9

Dealing with slider joints: (a) A slider joint between links i and j, (b) adjusting a box to fit a hyperbolic paraboloid zi=xili, and (c) the tetrahedron defined by the Pi′’s is a convex bound of this surface inside Bc″




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