%% 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{An omnidirectional wheel based on Reuleaux-triangles}

\author{deleted for blind review}
%Jochen Brunhorn}

\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 at the same distance from the
center of rotation. 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.
\end{abstract}




\section{Omnidirectional wheels}

Omnidirectional wheels have been used in robotics, in industry,
and in logistics for many years. The main source for
omnidirectional wheels are  companies which produce them for
omnidirectional transportation tables, such as those used by
postal or package handling companies. Omnidirectional wheels have
been used to build omnidirectional robots, specially in the
RoboCup setting. An omnidirectional robot can drive along a
straight line from point to point, while rotating along this line
in order to arrive with the correct orientation. Omnidrectional
wheels have also been used for wheel chairs, for service vehicles
in airports, and many other applications.

The first omnidirectional wheels were developed by the Swedish
inventor Bengt Ilon around 1973 \cite{bengt}. Fig.~\ref{ilon}
shows the design of the wheel and a fork lift built using them.
The profile of the wheel is more or less round. The wheel is
omnidirectional but transversal forces produce excessive friction
in the axes of the small rollers. A clever alternative are
``Killough rollers'' \cite{killough},  which are usually built
using two truncated spheres. Such rollers were used by the Cornell
RoboCup team in 2000, and were still in use until 2004 in RoboCup
competitions. Many RoboCup teams use now self-built wheels or a
variation of commercial wheels built for transport platforms, such
as those shown in Fig.~\ref{double}. Geometrically, such wheels
are polygons with small transversal rollers at the polygon's
corners (as can be seen in the left wheel). The wheels are made
round by using a stack of two of them and avoiding any gap in the
 wheel profile (right wheel in Fig.~\ref{double}). The point of contact changes from one
wheel to the other as the complete wheel rolls.

\begin{center}
\begin{figure}[thb]
\centering
\includegraphics[scale=0.55]{gif/bengtilon.eps}
\includegraphics[scale=0.55]{gif/gabelstapler.eps}
\caption{Omnidirectional wheel and omnidirectional fork
lift}\label{ilon}
\end{figure}
\end{center}



\begin{center}
\begin{figure}[thb]
\centering
\includegraphics[scale=0.15]{gif/rad.eps}
\includegraphics[scale=0.45]{gif/double.eps}
\caption{Segmented omnidirectional wheel and a commercial double
wheel }\label{double}
\end{figure}
\end{center}

Much effort has been spent on improving the Swedish or Mecanum
wheels, as they are sometimes called \cite{mecanum}. The
omniwheels can only roll smoothly if the profile of the complete
wheel assembly is perfectly round, without gaps. However, as we
show in the next sections, it is possible to build an
omnidirectional wheel with transversal rollers  with gaps between
the rollers which is nevertheless able to drive smoothly.


\section{Reuleaux triangles}


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 this
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}).
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}). 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. 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 $x=0$) to 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 appendices.

\begin{center}
\begin{figure}[thb]
\centering
\includegraphics[scale=0.6]{Figure3.eps}
  \caption{Profile of the Reuleaux wheel}\label{profile}
\end{figure}
\end{center}

The appendices 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 mentioned 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.


\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 half-wheel 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.

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}.

\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}

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. Conventional omnidirectional wheels are shaped as
$n$-side polygons, which leads to a movement where the robot is
lifted up every time a passive wheel touches the ground (see
Fig.~\ref{double}, left diagram). At high velocities, the entire
wheel might lose ground contact, making it difficult to drive
accurately (as happens during RoboCup competitions, \cite{simon}).

\begin{center}
\begin{figure}[thb]
\centering
\includegraphics[scale=0.25]{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}\label{third}
\end{figure}
\end{center}


\section{Design of excenter and gears}

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

Additionally, for each complete rotation around the excenter, the
Reulaux disc itself needs to rotate by $-\frac{2}{3}\pi$ around
its center. This combined movement is realized by two connected
gear wheels. The first gear wheel, with the sprockets on the
outside, is fixed to the first Reuleaux disc. Its diameter must be
$3\cdot d_{u}$. Since this gear wheels moves around the main axis
excentrically, a counterweight may be attached to the excenter to
compensate the gear wheel's centrifugal force. The second gear
wheel, with its sprockets on the inside, is fixed to  the robot.
Its diameter is  be $4\cdot d_{u}$. Figure 6 shows the complete
configuration.

\begin{center}
\begin{figure}[thb]
\centering
\includegraphics[scale=0.7]{Figure6.eps}
  \caption{Gear wheels and their diameter}\label{fourth}
\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. The phase shift is chosen in such way
that the first disc's center travels around the displacement curve
through $\pi$ ahead, with the discs rotated around their centers
by $\frac{1}{3}\pi$ to each other (this is explained above by the
relation between both movements). Building a gear for this task is
rather simple: the second disc's center axis is placed at distance
$\frac{1}{2}d_{u}$ to the main axis, in the opposite direction of
the first disc's axis. For the rotation of the second disc itself,
the gear is constructed to provide a 1:1 translation from the
first to the second disc.

\begin{center}
\begin{figure}[thb]
\centering
\includegraphics[%
  scale=0.7]{Figure7.eps}
    \caption{Assembly and movement of the two Reuleaux discs}\label{fifth}
  \end{figure}
\end{center}


\begin{center}
\begin{figure}[thb]
\centering
\includegraphics[scale=0.3]{gif/Modell_Abstand.eps}
    \caption{CAD model of the complete Reuleaux omnidirectional wheel assembly}\label{sixth}
\end{figure}
\end{center}


\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. 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.

Fig.~\ref{sixth} shows a ready-to-build sample design, including
all the components described above. The wheel has been simulated
by animation in a CAD system. A prototype of this wheel will be
exhibited at the RoboCup 2006 competition in Bremen. It is a new
attempt at reinventing the wheel \cite{reinventing}, right, but
the omnidirectional wheel.



\begin{thebibliography}{9}



\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{killough} S.M. Killough, and F.G. Pin, ``A New Family of
Omnidirectional and Holonomic Wheel Platforms for Mobile Robots'',
{\em IEEE Transactions on Robotics and Automation}, Vol. 10, N. 4,
1994, pp. 480-489.

\bibitem{mecanum} O. Diegel, A. Badve, G. Bright, J. Potgieter, S.
Tlale, ``Improved Mecanum Wheel Design for Omni-directional
Robots'', Proc. 2002 Australasian Conference on Robotics and
Automation, Auckland, November 2002, pp. 117-121.

\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.

\bibitem{reinventing} J. C. Bongard, and H. Lipson, ``Reinventing
the Wheel: An Experiment in Evolutionary geometry'', GECCO 2005,
Washington, 2005, pp.

\end{thebibliography}

\end{document}
