Wheelie Directory Reference

Detailed Description

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

Introduction

All members of the Wheelie family have two particularities:

• They cannot be solved in closed form.
• Their position analysis cannot be decomposed into simpler problems. In other words, it cannot be solved by merging the solutions of sub-linkage modules.

The first property derives from the fact that the linkage graphs are planar and triconnected and, by the Owen-Power theorem, this implies their solution cannot be found by ruler-and-compass methods [Owen 2007]. The second property can be proved by showing that such graphs only contain a single triconnected component. Thus, solvers based on ruler-and-compass methods or on modular kinematics cannot solve such linkages. Therefore, the Wheelie linkages are a perfect scalable testbed for general position analysis algorithms.

Geometry

As it can be seen in the figure above, a T-Wheelie linkage is formed by a loop of T equally sized isosceles triangles generated from two concentric circles and where the triangles are connected with the following pattern:

• Two consecutive triangles around the circle are connected by two bars.
• Two triangles with one triangle in between are directly connected by one bar.
• The linkage is made rigig adding (T-6)/2 bars, creating (T-6)/2 adjacent triangles in the interior circle.

Process

Please, see the individual files for the instructions on how to process them:

• Wheelie6 A wheelie with 6 triangles.
• Wheelie8 A wheelie with 8 triangles.
• Wheelie10 A wheelie with 10 triangles.
• Wheelie6mobile A wheelie with 6 triangles where one of the bars is removed. With this, the configuration space is one-dimensional instead of zero-dimensional.

Statistics

Characteristics of the problem:

	Wheelie6	Wheelie8	Wheelie10	Wheelie6Mobile
Nr. of indep. loops	10	14	18	9
Nr. of links	21	29	37	20
Nr. of joints	18	24	30	18
Nr. of equations (in the simplified system)	40	56	72	37
Nr. of variables (in the simplified system)	40	56	72	38

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

	Wheelie6	Wheelie8	Wheelie10	Wheelie6Mobile
Nr. of Empty boxes	1485	4135	12671	5867
Nr. of Solution boxes	19	17	25	2595
Execution time (s)	300	1800	7600	1900

The execution times can be significantly reduced executing cuik on a cluster of computers. This is easy using the rmpicuik script (assuming that you have a properly configured cluster of computers).

Results

These are the 19 valid configuraions of the Wheelie6.world problem:

This is a projection of the one-dimensional configuration space of the Wheelie6mobile.world problem.

Actually, this configuration space contains 16 connected components, corresponding to 16 different assembly modes of the linkage. One of such modes is rigid, and the remaining ones are mobile. The mobile modes show up as disjoint closed curves in the above plot (they are better appreciated in the 3d plot, as seen with geomview).

References

• J. C. Owen , S. C. Power, "The non-solvability by radicals of generic 3-connected planar graphs", Transactions of the American Mathematical Society no. 359, pp. 2269-2303, 2007.

Files
file	Wheelie10.world [code]
	The 10-Wheelie mechanism.
file	Wheelie6.world [code]
	The 6-Wheelie mechanism.
file	Wheelie6mobile.world [code]
	A modified 6-Wheelie mechanism with one degree of freedom.
file	Wheelie8.world [code]
	The 8-Wheelie mechanism.