%% LyX 1.3 created this file.  For more info, see http://www.lyx.org/.
%% Do not edit unless you really know what you are doing.
\documentclass{llncs}
\usepackage{llncsdoc}
\usepackage{graphicx}
\usepackage{psfig}
\usepackage{epstopdf}
\parindent0mm
\parskip5mm

%\usepackage[T1]{fontenc}
%\usepackage[latin1]{inputenc}


%\makeatletter

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LyX specific LaTeX commands.
%% Bold symbol macro for standard LaTeX users
\newcommand{\boldsymbol}[1]{\mbox{\boldmath $#1$}}


%\usepackage{babel}
%\makeatother
\title{A novel omnidirectional wheel based on Reuleaux-triangles}

\author{Jochen Brunhorn, Oliver Tenchio, Ra{\'u}l Rojas}
\institute{Institut f{\"u}r Informatik\\Freie Universit{\"a}t
Berlin\\Takustr. 9, 14195 Berlin, Germany}

\begin{document}
\markboth{\LaTeXe{} Class for Lecture Notes in Computer
Science}{\LaTeXe{} Class for Lecture Notes in Computer Science}
\thispagestyle{empty}


\maketitle

\begin{abstract}
This paper discusses the mechanical design and simulation of a
novel  omnidirectional wheel based on Reuleaux-triangles. The main
feature of our omniwheel is that  the point of contact of the
wheel with the floor is always kept at the same distance from the
center of rotation by mechanical means. This produces smooth
translational movement on a flat surface, even when the profile of
the complete wheel assembly has gaps between the passive rollers.
The grip of the wheel with the floor is also improved. The design
described in this paper is ideal for hard surfaces, and can be
scaled to fit small or large vehicles. This is the first design
for an omnidirectional wheel without circular profile, yet capable
of rolling smoothly on a hard surface.
\end{abstract}




\section{Motivation and Reuleaux triangles}


It has been thought for many years, that the only way of providing
a smooth rolling effect when using omnidirectional wheels\footnote{The very first omnidirectional wheel
was patented in 1919 by J. Grabowiecki in the  US \cite{grabo}. Bengt Ilon patented another in 1973 \cite{bengt}.} with
gaps between rollers is: a) by stacking two wheels on the same
axis, producing a combined circular profile; b) by using several
synchronized wheels which combine in order to support the vehicle
 keeping the distance to the floor constant; or c) by using spheres or quasi-spheres as
wheels. There is a fourth alternative, which is to design
omniwheels with a circular profile, in which the gaps are almost
closed by using two kinds of rollers alternatively, as in
\cite{korea}. Long and short rollers alternate on the periphery of
the wheel. In this paper we show for the first time that it is
possible to build an omnidirectional wheel {\em with gaps} between
the transversal rollers, that is without circular profile,  which
is nevertheless able to drive smoothly.


A so-called Reuleaux triangle (named after the German engineer
Franz Reuleaux, who was a professor of mechanical engineering at
the Technical University of Berlin) is a geometric shape whose
width remains constant during rotation. This means that two
parallel lines in contact with the shape's boundary stay at the
same distance independently of the shape's orientation. The
simplest shape of constant width is a circle. Other geometric
figures can be modified to have constant width.
Fig.~\ref{triangle} shows how a Reuleaux triangle is constructed.
Starting from an equilateral triangle of side length $l$, constant
width is achieved  adding circular arcs with radius $l$ to each of
the triangle's sides. The center of each arc is placed at the
corner opposite the respective side. Reuleaux first mentioned
these triangles in 1876 \cite{reuleaux}.

\begin{center}
\begin{figure}[thb]
\centering
\includegraphics[scale=0.55]{Figure1.eps}
\caption{Construction of the Reuleaux triangle}\label{triangle}
\end{figure}
\end{center}

The shape's area can be increased by extending the triangle's
sides beyond the corners by a distance $x$ (Fig~\ref{triangle},
right). The arcs' radii become then  $l+x$. The gaps  outside the
original triangle and between two crossing sides are closed by
circular arcs of radius $x$. In the following, the smaller arcs of
radius $x$ will be referred to as $l_{x}$-sections, and the larger
arcs as $l_{r}$-sections.

\begin{center}
\begin{figure}[thb]
\centering
\includegraphics[scale=0.7]{Figure2.eps} \caption{The Reuleaux
triangle rotating inside a square}\label{arcs}
\end{figure}
\end{center}

The  characteristics of a Reuleaux triangle allows such shape to
rotate inside a square, as a circle also can (Fig.~\ref{arcs}, see
\cite{weisstein} for an explanation). Notice that the triangle's
center \emph{does not} remain in the same place while rotating the
triangle, but moves along a curve consisting of four elliptical
arcs. For the wheel design presented in this paper, this curve is
approximated by a circle\footnote{The  curve is  nearly the
superellipse $x^{2.36}+y^{2.36}=c$, where $c$ is a constant
\cite{weisstein}.}. The circle's diameter $d_{u}$ is obtained
using the following formula:\[
d_{u}=l\cdot(\frac{4}{3}\cdot\cos(\pi/6)-1)\] obtained from
elementary geometrical considerations. This diameter \emph{remains
the same} for a given $l$, independent of the chosen extension of
length $x$. Therefore, the principal movement can be obtained from
considering simple Reuleaux-triangles (that is, triangles for
which $x=0$) instead of extended triangles. This fact is important
since the wheel design presented here makes use of extended
Reuleaux-triangles.




\section{Wheel design}

The omnidirectional wheel proposed in this paper consists
basically of the following parts:
\begin{enumerate}
\item Two discs based on Reuleaux-triangles, each carrying three passive
wheels.
\item A gear connecting these discs, which allows the transmission of rotation between
both.
\item An excenter which holds both discs and the gear.
\end{enumerate}
%\subsection{Reuleaux discs}
In our wheel, a Reuleaux-triangle of enhanced area is used for the
shape of the component which carries each group of three passive
wheels (Fig.~\ref{profile}). The shape of the passive wheels'
profile is determined by
  the $l_{x}$-sections described above.
The $l_{r}$-sections remain empty, except for small supporting
structures.

The structures do not allow the passive wheels to lose contact
with the ground as long as the next passive wheel has not reached
the ground yet. The path described by such passive wheels is
explained below.

The passive wheel is built by using the $l_x$-section shape as the
profile of a roller. Note that the position of the center axis of
this roller (which is actually the passive wheel's axis) may be
chosen arbitrarily. Thus it is possible to construct passive
wheels of different sizes.
\begin{center}
\begin{figure}[thb]
\centering
\includegraphics[scale=0.6]{Figure3.eps}
  \caption{Profile of the Reuleaux wheel (left). The wheel
  has three passive rollers. The rollers can have a smaller radius (centerline $A$),
  or a larger radius (centerline $B$). In each case, the effect is the same.}\label{profile}
\end{figure}
\end{center}
%\subsection{The wheel's movement}
Before describing the remaining parts, it is necessary that the
reader  understands the principle behind the functionality of this
omnidirectional wheel.

As described above, a Reuleaux-triangle can be rotated inside a
square \cite{weisstein}. The square's sides are touched
alternately by the triangle's corners and the circular arcs (in
the extendend triangle by the $l_{x}$-section and the
$l_{r}$-section). Imagine one of the square's sides to be the
ground over which the
 Reuleaux discs roll, with the passive wheels touching
the ground (remember that the passive wheels' profile matches the
curve that describes the $l_{x}$-section). Each disc has contact
with the ground only part of the time, because when the
$l_{x}$-section loses contact with one side, the $l_{r}$-section
reaches that side. Remember that the $l_{r}$-section-surface is
\emph{empty} in the Reuleaux discs.

\begin{center}
\begin{figure}[thb]
\centering
\includegraphics[scale=0.7]{Figure4.eps}
  \caption{Two Reuleaux discs shifted in phase}\label{second}
\end{figure}
\end{center}
To guarantee ground contact the entire time, a second Reuleaux
disc is added to the wheel design, describing the exact same path
of movement, but with a phase shift. The phase shift corresponds
to exactly one half of a rotation of the Reuleaux-triangles'
center around the displacement path of diameter $d_{u}$, as
illustrated in Fig.~\ref{second} (one triangle looks as
``flipped'' by vertical reflection ).

\begin{center}
\begin{figure}[thb]
\centering
\includegraphics[scale=0.20]{gif/gesamt.eps}
  \caption{Rotation of the wheel and relative displacement of the rollers.
    The wheel starts on the upper left and rolls counterclockwise.
    The passive roller on the ground loses contact only when the next roller
    has reached the ground-contact position. The combined movement looks as if the
    next roller ``stretches a leg'' before reaching  ground.}\label{third}
\end{figure}
\end{center}

This ensures that the Reuleaux discs' passive wheels always touch
the ground, alternating from one disc to the other. A simulation
of a working omnidirectional wheel following this principle is
shown in Fig.~\ref{third}. The advantage of this wheel design
becomes clear immediately: The passive wheels describe a purely
linear path along the ground, preventing any up- and
down-movement. On the contrary, since conventional omnidirectional
wheels are shaped as $n$-side polygons (see Fig.~\ref{double},
left diagram), this leads to an up-and-down movement of the robot,
which is lifted up every time a passive wheel touches the ground .
At high velocities, the entire wheel might lose ground contact,
making it difficult to drive accurately (as often happens during
RoboCup competitions.


\begin{center}
\begin{figure}[thb]
\centering
\includegraphics[scale=0.14]{gif/onegear.eps}
  \caption{Gear wheels and their diameter. When rotating around the
  gear fixed to the robot, the center of the Reuleaux disc describes
  a circle of radius $d_u$.}\label{fourth}
\end{figure}
\end{center}

\section{Design of excenter and gears}

In order to achieve the circular movement of the center of each
Reuleaux disc an excenter is needed. This excenter is held in
place by the motor axis to which the torque is applied in order to
drive the omnidirectional wheel. The first Reuleaux disc center
axis is placed at a distance $\frac{1}{2}d_{u}$ of the motor axis
(black rod in Fig.~\ref{fourth}). When the excenter is rotated,
the Reuleaux disc's center describes the desired curve (a circle
of diameter $d_{u}$).

Additionally, for each complete rotation around the excenter, the
Reuleaux disc itself needs to rotate by $-\frac{2}{3}\pi$ around
its center. This combined movement is realized by two connected
gear wheels. The  smaller gear wheel, with the sprockets on the
outside, is fixed to the  Reuleaux disc. Its diameter must be
$3\cdot d_{u}$. Since this gear wheel rotates around the motor
axis excentrically, a counterweight may be attached to the
excenter to compensate the gear wheel's centrifugal force. The
second larger gear wheel, with its sprockets on the inside, is
fixed to the robot. Its diameter is $4\cdot d_{u}$.
Fig.~\ref{fourth} shows the complete configuration.

\begin{center}
\begin{figure}[thb]
\centering
\includegraphics[scale=0.6]{Figure7.eps}
\includegraphics[scale=0.14]{gif/combined.eps}
%\includegraphics[scale=0.11]{gif/doublegear.eps}
%\includegraphics[scale=0.19]{gif/twogears.eps}
\caption{Assembly and movement of the two Reuleaux discs. To the
left a diagram, in the middle a cross section of the assembly, and
to the right the transmission gear.}\label{fifth}
  \end{figure}
\end{center}





Now that the first Reuleaux disc describes the desired movement,
the second disc needs to be connected to imitate the exact same
movement with a phase shift. To obtain a phase shift of
$\frac{1}{3}\pi$, we just flip the Reuleaux structure by 180
degrees, as shown in Fig.~\ref{fifth}. Building a gear for this
task is not difficult: the second disc's center axis is placed at
a distance $\frac{1}{2}d_{u}$ from the motor axis, as before, but
in the opposite direction of the first disc's axis. In order to
rotate the second disc, a transmission gear is constructed to
provide a 1:1 transmission of rotation from the first to the
second disc (Fig.~\ref{fifth}).

\section{Conclusions}

This paper has presented a new design for an omnidirectional wheel
based on Reuleaux triangles. We have shown that even when the
complete profile of the wheel mount has ``gaps'' between the
passive rollers, it is indeed possible to achieve smooth
rotational movement. For this to occur, the wheel center rotates
around an excenter. A conventional wheel with profile gaps
resembles a polygon and a rotating polygon moves the wheel's
center up and down -- if nothing is done against that movement.
The wheels used by almost all teams in the small-size league
resemble such polygons. Mid-size teams have used also wheels with
perpendicular passive rollers and gaps in the wheel profile. The
wheels presented in this paper are an alternative solution.


We are aware that the wheel design described in this paper is more
complex than other omnidirectional wheels and requires more
mechanical parts. Nevertheless, the wheel presented here has some
interesting theoretical properties. Firstly, its profile is not
circular, correcting the long held misconception that only
circular-profile omniwheels can roll smoothly. Secondly, this
design allows a robot (or other vehicle) highly precise and
controllable omnidirectional driving, even at high speeds. Wheel
grip is optimal because driving is vibration free, and the
individual passive wheels have contact with the ground over a long
time. Modelling Mecano wheels is extremely complex (because of the
angle at which the passive rollers are placed) \cite{kinematic}.
Kinematic modeling of the Reuleaux wheels is much simpler.

\begin{center}
\begin{figure}[thb]
\centering
%\includegraphics[scale=0.16]{gif/Modell_Abstand.eps}
\includegraphics[scale=0.07]{radnew.ps}
\includegraphics[scale=0.088]{foto_omnirad.eps}

    \caption{Our first two Reuleaux omnidirectional wheel prototypes}\label{sixth}
\end{figure}
\end{center}




Fig.~\ref{sixth} shows a ready-to-build sample design, including
all the components described above and a photograph of the first prototype. 
The smooth rolling movement was
validated in the computer before the prototype was built.  Two Reuleaux 
omnidirectional 
wheels will be on display at the
RoboCup 2006 competition in Bremen. It is a new attempt at
reinventing the wheel, right, but the
omnidirectional wheel.



\begin{thebibliography}{9}


\bibitem{grabo} J. Grabowiecki, ``Vehicle Wheel'', US Patent
1,303,535, June 3, 1919.

\bibitem{bengt} B. E. Ilon, ``Wheels for a course stable
self-propelling vehicle movable in any desired direction on the
ground or some other base'', 1975, US Patent 3,876,255.

\bibitem{korea} K-S. Byun, and J-B. Song, ``Design and
Construction of Continuous Alternate Wheels for an Omnidirectional
Mobile Robot'', {\em Journal of Robotic Systems}, Vol. 9, N. 9,
pp. 569--579.

\bibitem{reuleaux} F. Reuleaux, {\em The Kinematics of Machinery:
Outlines of a Theory of Machines}, MacMillan, London, 1876.

\bibitem{weisstein} E. W. Weisstein, ``Reuleaux Traingle'', from
MathWorld - A Wolfram Web Resource,
mathword.wolfram.com/ReleauxTriangle.html.

%\bibitem{simon} deleted for blind review

\bibitem{kinematic} P. F. Muir, and Ch. P. Neuman, ``Kinematic
Modeling for Feedback Control of an Omnidirectional Wheeled Mobile
Robot'', in I. J. Cox, and G. T. Wilfong (eds.), {\em Autonomous
Robot Vehicles}, Springer-verlag, Berlin, 1990.



\end{thebibliography}

\end{document}
