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\title{A novel omnidirectional wheel based on Reuleaux-triangles}

\author{Jochen Brunhorn, Oliver Tenchio, Raul Rojas}
%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 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{A short history of omnidirectional wheels}

Omnidirectional wheels have been used in robotics, in industry,
and in logistics for many years. The main source of
omnidirectional wheels are  companies which produce them for
omnidirectional conveyor systems, for example, for handling
packages. Omnidirectional wheels are popular for omnidirectional
robots, specially in the RoboCup setting. An omnidirectional robot
can drive along a straight line from point $A$ to point $B$, while
rotating along the line in order to arrive with the correct
orientation. Omnidrectional wheels have also been used for
wheelchairs, for service vehicles in airports, and many other
applications.

It is not widely known that the very first omnidirectional wheel
was patented in 1919 by J. Grabowiecki in the  US \cite{grabo}.
Fig.~\ref{grabowheel} shows an image from the patent application.
The assembly consists of a main wheel and transversal rollers,
such as those used by most RoboCup teams.  As early as 1907,
inventors were considering the design of vehicles capable of
moving forward and sideways without steering the wheels
\cite{kennedy}.

\begin{center}
\begin{figure}[thb]
\centering
\includegraphics[scale=0.17]{grabo.eps}
\caption{Drawing of probably the first omnidirectional wheel, as
described in Grabowiecki's US Patent of 1919}\label{grabowheel}
\end{figure}
\end{center}

One of the first modern omnidirectional wheels was developed by
the Swedish inventor Bengt Ilon around 1973 \cite{bengt}.
Fig.~\ref{ilon} shows the design of the Ilon wheel and a fork lift
built using them. The profile of the wheel is very nearly
circular. 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. Although the rollers are named
after Killough, the rollers had been actually patented in 1980 by
Bradbury \cite{bradbury}.
\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}
Fig.~\ref{rollers} shows an example of the kind of vehicles that
can be built with the Killough wheels (left), and with a double
omnidirectional wheel, each with only two rollers, in order to
provide a circular profile (Fig.~\ref{rollers}, right).

\begin{center}
\begin{figure}[thb]
\centering
\includegraphics[scale=0.22]{gif/roller.eps}
\caption{Two  vehicles with omnidirectional wheels, as in
\cite{bradbury} }\label{rollers}
\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}
 Many
RoboCup teams use now self-built wheels or a variation of
commercial wheels  for conveyor systems, 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). Blumrich managed to patent similar wheels in
1974, although they are just a simple variation of the Grabowiecki
wheels \cite{blumrich}. 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. A theoretical
analysis of omnidirectional wheels with transversal rollers and
gaps can be found in \cite{muir}.




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. Therefore, some
groups have used spheres as a basis for the robot. The spheres can
be activated with  rollers, as in a mechanical mouse \cite{endo},
or a group of spheres can be moved by chains or transversal bars
\cite{damoto},\cite{asada}. The spheres provide smooth rolling but
the necessary mechanics is rather bulky. Conventional wheels can
be also used for omnidirectional robots, if the wheels are rotated
by a second steering motor \cite{wada}.


It has been thought for many years, that the only way of providing
a smooth rolling effect when using omnidirectional wheels 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 (as in
Fig.~\ref{rollers}); or c) by using spheres or quasi-spheres as
wheels (in the movie iRobot, for example, cars are omnidirectional
and roll on spheres). 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. The result
is primarily of theoretical interest and will hopefully open new
avenues of research around the time-honored subject of
omnidirectional wheels.


\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
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.



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.
\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}

\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}

\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}
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.


%\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 ).



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, \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. The combined movement looks as if the
    next roller ``stretches a leg'' before reaching  ground.}\label{third}
\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=0.17]{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}

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}).

\begin{center}
\begin{figure}[thb]
\centering
\includegraphics[scale=0.7]{Figure7.eps}
\includegraphics[scale=0.22]{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}

\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. 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.

Fig.~\ref{sixth} shows a ready-to-build sample design, including
all the components described above. The wheel has been simulated
and animated in a CAD system. The smooth rolling movement has been
validated.  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.



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\bibitem{simon} deleted for blind review

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\end{thebibliography}

\end{document}
