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Research Papers

Cusp Points in the Parameter Space of Degenerate 3-RP R Planar Parallel Manipulators

[+] Author and Article Information
Montserrat Manubens

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

Guillaume Moroz

INRIA Nancy-Grand Est, 615, rue du jardin botanique, 54600, Villers-lès-Nancy, Franceguillaume.moroz@inria.fr

Damien Chablat

Institut de Recherche en Communicationset Cybernétique de Nantes, UMR CNRS n 6597, 1 rue de la Noë, 44321 Nantes, Francedamien.chablat@irccyn.ec-nantes.fr

Philippe Wenger

Institut de Recherche en Communicationset Cybernétique de Nantes, UMR CNRS n 6597, 1 rue de la Noë, 44321 Nantes, Francephilippe.wenger@irccyn.ec-nantes.fr

Fabrice Rouillier

INRIA Paris-Rocquencourt,  Université Pierre et Marie Curie Paris VI 4, place Jussieu, F-75005 Paris, Francefabrice.rouillier@inria.fr

J. Mechanisms Robotics 4(4), 041003 (Aug 10, 2012) (8 pages) doi:10.1115/1.4006921 History: Received May 30, 2011; Revised April 23, 2012; Published August 10, 2012; Online August 10, 2012

This paper investigates the conditions in the design parameter space for the existence and distribution of the cusp locus for planar parallel manipulators. Cusp points make possible nonsingular assembly-mode changing motion, which increases the maximum singularity-free workspace. An accurate algorithm for the determination is proposed amending some imprecisions done by previous existing algorithms. This is combined with methods of cylindric algebraic decomposition, Gröbner bases, and discriminant varieties in order to partition the parameter space into cells with constant number of cusp points. These algorithms will allow us to classify a family of degenerate 3-RP R manipulators.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 1

Example of degenerate 3-RP¯R

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Figure 2

Cusp point κ as a triple root of the FKP and non-singular path linking upper and lower solutions of the FKP

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Figure 3

Singular curve for ρ1=13 on (ρ2 ,ρ3 )

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Figure 4

Comparison of both discussions on ρ1

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Figure 5

Plot of the DV of CF with respect to (ρ1 ,d1 )

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Figure 6

Cell Decomposition for (ρ1 ,d1 )

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Figure 7

Zoom in of the cell decomposition for (ρ1 ,d1 ). Line d1  = 1 in white

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Figure 8

Distribution of cusp points on DV (CF ;ρ1 ,d1 ) (a), and zoom in view on [0,1.5] × [0,1.5] (b)

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Figure 9

Complete analysis of the cusp points for (ρ1 ,d1 )

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Figure 10

Optimal d1 and Δρ1 for noncuspidal degenerate 3-RP¯R

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