Research Papers

Application of Distance Geometry to Tracing Coupler Curves of Pin-Jointed Linkages1

[+] Author and Article Information
Nicolás Rojas

e-mail: nrojas@iri.upc.edu

Federico Thomas

e-mail: fthomas@iri.upc.edu
Institut de Robòtica i Informàtica
Industrial (CSIC-UPC),
Llorens Artigas 4-6,
Barcelona 08028, Spain

Some of the ideas contained in this paper were already presented at the ASME 2011 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference IDETC/CIE 2011, August 28–31, 2011, Washington, DC.

2Corresponding author.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the Journal of Mechanisms and Robotics. Manuscript received July 25, 2011; final manuscript received January 23, 2013; published online March 26, 2013. Assoc. Editor: Qiaode Jeffrey Ge.

J. Mechanisms Robotics 5(2), 021001 (Mar 26, 2013) (9 pages) Paper No: JMR-11-1082; doi: 10.1115/1.4023515 History: Received July 25, 2011; Revised January 23, 2013

In general, high-order coupler curves of single-degree-of-freedom plane linkages cannot be properly traced by standard predictor–corrector algorithms due to drifting problems and the presence of singularities. Instead of focusing on finding better algorithms for tracing curves, a simple method that first traces the configuration space of planar linkages in a distance space and then maps it onto the mechanism workspace, to obtained the desired coupler curves, is proposed. Tracing the configuration space of a linkage in the proposed distance space is simple because the equation that implicitly defines this space can be straightforwardly obtained from a sequence of bilaterations, and the configuration space embedded in this distance space naturally decomposes into components corresponding to different combinations of signs for the oriented areas of the triangles involved in the bilaterations. The advantages of this two-step method are exemplified by tracing the coupler curves of a double butterfly linkage.

Copyright © 2013 by ASME
Topics: Linkages , Geometry
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Grahic Jump Location
Fig. 5

(a) In the strip of triangles {ΔP1P10P2,ΔP2P10P4,ΔP10P3P4},s1,3 can be obtained from bilaterations. (b) After affixing the strip of triangles {ΔP1P6P3,ΔP1P5P6,ΔP6P5P9} to the previous one, s4,9 can also be obtained using bilaterations. (c) Likewise, s2,6 can be obtained after affixing the strip of triangles {ΔP4P9P7,ΔP7P9P8}. (d) A double butterfly linkage. (e) If the lengths of dotted segments were known, this double butterfly linkage would be equivalent to the obtained structure resulting from attaching three strips of triangles.

Grahic Jump Location
Fig. 4

Top: The bilateration problem. Bottom: Concatenation of two bilaterations.

Grahic Jump Location
Fig. 3

(a) Using the standard vector loop formulation, the configuration space of a double butterfly linkage can be represented by a one-dimensional variety in the space defined by {θ1,…,θ7}. (b) Alternatively, using the proposed approach, this configuration space can be represented by a one-dimensional variety in the space defined by {s1,6,s2,4} which can be decomposed into 16 components, one for each combination of signs of the oriented areas of the triangles ΔP2P4P10,ΔP1P3P6,ΔP1P6P5, and ΔP4P9P7.

Grahic Jump Location
Fig. 2

(a) A four-bar linkage and (b) the coupler curve traced by a point affixed to one of its bars while taking the opposite one as fixed. (c) Any coupler curve generated by this linkage can be expressed in terms of its configuration space which can be represented by a one-dimensional variety in the space defined by {θ1,θ2,θ3}, or by {θ1,θ2} if the distance constraint between P3 and P4 is used as closure condition instead of the standard loop equation. (d) Alternatively, this configuration space can be represented by value ranges of a single variable, s2,3, one range for each combination of signs of the oriented areas of the triangles ΔP1P2P3 and ΔP2P4P3.

Grahic Jump Location
Fig. 1

(a) In the predictor step, a point p* in the tangent line to the curve at the current point pi is estimated. (b) In the corrector step, the predicted point p* is adjusted onto the curve producing a new point pi+1. (c) The drifting problem. (d) The cycling problem.

Grahic Jump Location
Fig. 6

Top: The real solution set of Eq. (13) in the plane defined by s2,4 and s1,6 for sampled values of s2,4. Bottom: From left to right, the connected components of the configuration space traced when starting from the initial configurations s2,4=74, s1,6=188.68, and A2,4,10 > 0,A1,3,6 < 0,A1,6,5 < 0, and A4,9,7 < 0,s2,4=74,s1,6=122, and A2,4,10 > 0,A1,3,6 > 0,A1,6,5 > 0, and A4,9,7 < 0, and s2,4=74,s1,6=98.92, and A2,4,10 < 0,A1,3,6 < 0,A1,6,5 < 0, and A4,9,7 < 0, respectively.

Grahic Jump Location
Fig. 7

The paths followed by the revolute pair center P9 from different initial configurations. Top: For the curve traced from the initial configuration s2,4=74, s1,6=188.68, and A2,4,10 > 0,A1,3,6 < 0,A1,6,5 < 0, and A4,9,7 < 0, zoomed-in areas show how, after mapping the configuration space onto the workspace, a ramphoid cusp, and near-quadruple point are generated. Center: For the curve traced from the initial configuration s2,4=74, s1,6=122, and A2,4,10 > 0,A1,3,6 > 0,A1,6,5 > 0, and A4,9,7 < 0, a cusp and a tacnode can be identified. Bottom: For the curve traced from the initial configuration s2,4=74,s1,6=98.92, and A2,4,10< 0,A1,3,6 < 0,A1,6,5 < 0, and A4,9,7 < 0, zoomed-in areas show how, after mapping the configuration space onto the workspace, several singular points are generated in a reduced region.




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