DoubleButterfly Directory Reference

Detailed Description

[large]
Animation: 1, 2, 3, 4.


[Introduction] [Geometry] [Process] [Statistics] [Results] [References]

Introduction

The Double Butterfly Linkage is the one of the few eight-bar planar linkages that does not contain a four-bar loop. This indicates that the linkage cannot be decomposed into simpler, independently solvable parts. Using a counting theorem like Laman's [Laman, 1970], it can be shown that this mechanism moves with one internal degree of freedom. Such freedom may manifest in several assembly modes, as shown in the figure above.

The linkage becomes rigid if the orientation of one link (other than the ground) is held fixed, and in that case it can exhibit up to eighteen (rigid) assembly modes [Waldron, 1996].

This linkage is used in [Nielsen, 1999a] and [Wampler, 2001] to test the performance of resultant-based techniques. See [Porta, 2007] for further details.

Note that this simple mechanism can not be decomposed in active-pasive sub-parts and, thus, its path planning can not be addressed with the standard techniques to deal with closed-loops systems.

Geometry

The linkage is composed by 8 links: 4 triangles and 4 bars. The triangles are pair-wise articulated forming two butterflies. The following figure shows the arrangement of these 8 links as well as the reference systems used to formulate it.



The reference frames for the bars have the u vectors aligned with bar and its origin in the lower point of the bar. The v vector is the u vector rotated 90 degrees counter-clockwise. Due to the trivial relation between u and v vectors, only the components of the u vector w.r.t. the ground link appear in the final formulation.

As it can be seen in the figure, the linkage graph has three independent cycles that give rise to 6 loop equations (3 for the x components of the u vectors and 3 for the y components).

Note that the x and y components of the u vectors give the cosinus and sinus respectively of the orientation of the corresponding link w.r.t. the ground link. When directly formulating the linkage as a planer mechanisms (see [Porta, 2007]) the cosinus - sinus variables are used. When formulated as a spatial mechanism (see [Porta, 2008]) the system can be simplified until only the u_x and u_y variables for each link appear in the system.

Process

The instructions on how to process the examples are given in each separate file:

  • DoubleButterfly_fix.world Gives a formulation where the orientation of bar 1 is fixed (w.r.t. the ground link, t1). With this the solution space of the linkage becomes 0-dimensional.
  • DoubleButterfly.world Is a simple formulation that yields a 1-dof solution set

Statistics

Characteristics of the problem:

Fix Mobile
Nr. of indep. loops 3 3
Nr. of links 8 8
Nr. of joints 10 10
Nr. of equations (in the simplified system) 12 13
Nr. of variables (in the simplified system) 12 14

Here you have the statistics about the execution (on an Intel Core i7 at 2.9 Ghz).

Fix Mobile
Nr. of Empty boxes 0 34
Nr. of Solution boxes 6 623
Execution time (s) 0.25 25

Results

Here you have the 6 solutions of the fix butterfly.

Here you have an animation of the progressive refinement of the one-dimensional solution set:

This is the one-dimensional configuration space. This includes the four connected components (shown in different colors) whose animations can be seen using the links on the top of this page.

Below you have two different interactive LiveGraphics3D projections of this solution set. Drag inside the pictures to rotate the view; Shift-drag vertically to zoom. Note how relatively easy it is to distinguish the four assembly modes in the righthand-side projection, while in the left-side projection they appear totally intermingled.

References

  • J. M. Porta, L. Ros, and F. Thomas, "A Linear Relaxation Technique for the Position Analysis of Multi-Loop Linkages", Submitted to IEEE Transactions on Robotics, Available as a IRI Technical Report, 2008.
  • J. M. Porta, L. Ros, T. Creemers, and F. Thomas, "Box Approximations of Planar Linkage Configuration Spaces", ASME Journal of Mechanical Design, Vol. 129, pp. 397-405, April 2007.
  • T. Creemers, J. M. Porta, L. Ros, and F. Thomas, "Fast Multiresolutive Approximations of Planar Linkage Configuration Spaces", in Proc. IEEE International Conference on Robotics and Automation (ICRA), pp. 1511-1517, Orlando, Florida, 2006.
  • G. Laman, "On graphs and rigidity of plane skeletal structures", J. Engineering Math. 4, pp. 331-340, 1970.
  • K. J. Waldron and S. V. Sreenivasen, "A study of the position problem for multi-circuit mechanisms by way of example of the Double Butterfly linkage", ASME Journal of Mechanical Design, 1996, vol. 118, pp.390-395.
  • J. Nielsen and B. Roth, "Solving the Input/Output Problem for Planar Mechanisms", ASME Journal of Mechanical Design, June 1999, vol. 121, Nº 2, pp. 206-211.
  • C. W. Wampler, "Solving the Kinematics of Planar Mechanisms by Dixon Determinant and a Complex-Plane Formulation", ASME Journal of Mechanical Design, September 2001, vol. 123, pp. 382-387.

Files

file  DoubleButterfly.world [code]
 The Double Butterfly linkage: explicit formulation.
 
file  DoubleButterfly_fix.world [code]
 The Double Butterfly linkage: case with discrete solutions.