Research Papers

Closed-Form Solution to the Position Analysis of Watt–Baranov Trusses Using the Bilateration Method

[+] Author and Article Information
Nicolás Rojas

Federico Thomas

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

Some authors misspell it as Barranov.

J. Mechanisms Robotics 3(3), 031001 (Jul 01, 2011) (10 pages) doi:10.1115/1.4004031 History: Received December 03, 2010; Revised March 29, 2011; Published July 01, 2011; Online July 01, 2011

The exact position analysis of a planar mechanism reduces to compute the roots of its characteristic polynomial. Obtaining this polynomial almost invariably involves, as a first step, obtaining a system of equations derived from the independent kinematic loops of the mechanism. The use of kinematic loops to this end has seldom been questioned despite deriving the characteristic polynomial from them requires complex variable eliminations and, in most cases, trigonometric substitutions. As an alternative, the bilateration method has recently been used to obtain the characteristic polynomials of the three-loop Baranov trusses without relying on variable eliminations nor trigonometric substitutions and using no other tools than elementary algebra. This paper shows how this technique can be applied to members of a family of Baranov trusses resulting from the circular concatenation of the Watt mechanism irrespective of the resulting number of kinematic loops. To our knowledge, this is the first time that the characteristic polynomial of a Baranov truss with more that five loops has been obtained, and hence, its position analysis solved in closed form.

Copyright © 2011 by American Society of Mechanical Engineers
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Figure 1

Left column: the Stephenson linkage, the Stephenson pattern resulting from concatenating four Stephenson linkages, and the Stephenson–Baranov truss resulting from the circular concatenation of four Stephenson linkages. Right column: the Watt linkage, the Watt pattern resulting from concatenating four Watt linkages, and the Watt–Baranov truss resulting from the circular concatenation of four Watt linkages.

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Figure 2

The bilateration problem in R2

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Figure 3

The general n-link Watt–Baranov truss has k=(n-1)/(n-1)22 loops and v=3/322(n-1) revolute joints. pv-1,v can be expressed as a function of p1,3 by computing 3k-2 bilaterations.

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Figure 4

The assembly modes of the analyzed 11-link Watt–Baranov truss

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Figure 5

The assembly modes of the analyzed 13-link Watt–Baranov truss (Part 1)

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Figure 6

The assembly modes of the analyzed 13-link Watt–Baranov truss (Part 2)

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Figure 7

The assembly modes of the analyzed 13-link Watt–Baranov truss (Part 3)




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