Wheelie Directory Reference

The Wheelie family of mechanisms.

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.

Detailed Description

[Introduction][Geometry][Formulations][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.

Formulations

This directory includes the following Wheelie mechanisms:

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.

In all cases we use the implicit formulation (i.e., refering to the joint articulation points using the points defining the triangles) assuming that points 0 and 1 of the triangles are those on the external circle and point 2 is on the internal one.

References