sensors
Article
Design, Implementation and Validation of the
Three-Wheel Holonomic Motion System of the
Assistant Personal Robot (APR)
Javier Moreno 1 , Eduard Clotet 1 , Ruben Lupiañez 1 , Marcel Tresanchez 1 , Dani Martínez 1 ,
Tomàs Pallejà 2 , Jordi Casanovas 1 and Jordi Palacín 1, *
1
2
*
Department of Computer Science and Industrial Engineering, University of Lleida, 25001 Lleida, Spain;
jmoreno@diei.udl.cat (J.M.); eclotet@diei.udl.cat (E.C.); robotica@udl.cat (R.L.);
mtresanchez@diei.udl.cat (M.T.); dmartinez@diei.udl.cat (D.M.); jcasanovas@quimica.udl.cat (J.C.)
Barton Laboratory, Cornell University, Geneva, NY 14456, USA; tpc63@cornell.edu
Correspondence: palacin@diei.udl.cat; Tel.: +34-973-702-724
Academic Editor: Gonzalo Pajares Martinsanz
Received: 15 June 2016; Accepted: 29 September 2016; Published: 10 October 2016
Abstract: This paper presents the design, implementation and validation of the three-wheel
holonomic motion system of a mobile robot designed to operate in homes. The holonomic motion
system is described in terms of mechanical design and electronic control. The paper analyzes
the kinematics of the motion system and validates the estimation of the trajectory comparing the
displacement estimated with the internal odometry of the motors and the displacement estimated
with a SLAM procedure based on LIDAR information. Results obtained in different experiments
have shown a difference on less than 30 mm between the position estimated with the SLAM and
odometry, and a difference in the angular orientation of the mobile robot lower than 5◦ in absolute
displacements up to 1000 mm.
Keywords: holonomic motion; assistant robot; mobile robot motion; omnidirectional wheel
1. Introduction
The uses of mobile robots are continuously increasing in non-industrial applications such as
military and security settings [1], inspection of power lines in smart grids [2], crop-inspection in
smart agriculture [3], disaster recovery [4], interaction with customers [5] and also helping people
with mobility impairments [6]. Reports from diverse institutions such as the United Nations [7] and
World Health Organizations [8] postulate that the proportion of people aged 60 or more will rise from
12% to 21% during the next 35 years as a result of a clear increase of human life expectancy and the
development of assistant robots can be a technological tool that will contribute to increase the quality
of life of elderly people and people with mobility impairments. In this direction, the combination of
assistant mobile robots [9] and fixed domotic systems [10] can be used at home in an unstructured
domestic environment and also contribute to supervise or develop some domestic tasks.
In this direction, the Assistant Personal Robot (APR) [9] proposed the conception of a new
robotic assistant designed to operate in tight indoor spaces thanks to its holonomic motion system.
The new contribution of this paper is the complete description of the optimized mechanical design,
kinematics, and basic control of the three-wheel holonomic motion system implemented in the APR
mobile robot. This complete description is proposed in order to foster replication and verification of
the results. The paper ends with an empirical validation of the trajectory estimated by the internal
control system which will be used in the future to improve the implementation of additional trajectory
control procedures.
Sensors 2016, 16, 1658; doi:10.3390/s16101658
www.mdpi.com/journal/sensors
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2.2.Background
Background
Mobile
robots can be classified accordingly to the motion system or by the type of mobility
2. Background
2. Background
2.
Background
(Table
1).
The
motion
system
can be based
on wheels,
tracks,
ball-shaped
legs.
The
type of
2.Mobile
Background
bebeclassified
accordingly
totothe
motion
system
by
type
ofofmobility
Mobilerobots
robotscan
can
classified
accordingly
the
motion
systemororwheels
bythe
theor
type
mobility
Mobile
robots
can be
accordingly
to thetomotion
system
or byorthe
of mobility
Mobile
robots
canclassified
be classified
accordingly
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system
bytype
the type
of mobility
mobility
can
bemotion
classified
as be
omnidirectional
holonomic)
or
non-omnidirectional.
The
holonomic
(Table
system
can
bebebased
on
wheels,
tracks,
ball-shaped
or
The
type
(Table1).
1).The
The
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wheels,
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legs.
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Mobile
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classified
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the
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orwheels
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robots
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or
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theorof
type
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(Table
1).Mobile
The
system
can
be
based
on wheels,
tracks,
ball-shaped
wheels
or legs.
The of
type
of of
(Table
1).motion
The
motion
system
can
be based
on wheels,
tracks,
ball-shaped
wheels
legs.
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type
mobility
can
be
classified
as
(or
holonomic)
or
non-omnidirectional.
The
mobile
robots
have
the advantage
theyon
can
change
the
direction
of motion
without
having
to
(Table
1).
The
motion
system
canthat
be based
wheels,
tracks,
ball-shaped
wheels
or legs.
The
type
of
mobility
can
be
classified
asomnidirectional
omnidirectional
(or
holonomic)
or
non-omnidirectional.
Theholonomic
holonomic
(Table
1).
The
motion
system
can
be
based
on
wheels,
tracks,
ball-shaped
wheels
or
legs.
The
type
of
mobility
can be
classified
as omnidirectional
(or holonomic)
or non-omnidirectional.
The holonomic
mobility
can
bebeclassified
as omnidirectional
omnidirectional
holonomic)
or non-omnidirectional.
The holonomic
mobility
classified
as
(or(or
holonomic)
non-omnidirectional.
The
holonomic
mobile
robots
have
the
advantage
that
can
change
the
direction
ofofmotion
without
having
toto
mobile
robotscan
have
the
advantage
thatthey
can
change
theor
direction
motionfrom
without
having
perform
intermediate
rotation
steps
and
they
are
able
to
move
in
all
directions
a
given
starting
mobility
can
behave
classified
as omnidirectional
(or
holonomic)
ordirection
non-omnidirectional.
The
holonomic
mobile
robots
have
the
advantage
thatthat
they
cancan
change
the
direction
of
without
having
mobile
robots
the
advantage
thatthey
they
can
change
the
of motion
without
having
mobile
robots
have
the rotating
advantage
change
the
direction
of motion
motion
without
having
to to to
perform
intermediate
rotation
steps
ininallalldirections
from
starting
perform
intermediate
rotation
stepsand
andthey
theyare
areable
abletotomove
move
directions
froma agiven
given
starting
point
while
simultaneously
[11].
mobile
robots
have
the
advantage
that
they
can
change
the
direction
of
motion
without
having
to
perform
intermediate
rotation
steps
and
they
are
able
to
move
in
all
directions
from
a
given
starting
perform
intermediate
rotation
steps
and
they
are
able
to
move
in
all
directions
from
a
given
perform
intermediate rotation
steps
and they are able to move in all directions from a given startingstarting
point
rotating
[11].
pointwhile
whilesimultaneously
simultaneously
rotating
[11].
intermediate
rotation
steps
and
they
are
able
to
move
in
all
directions
from
a
given
starting
pointperform
while
simultaneously
rotating
[11].
point
while
simultaneously
rotating
[11].
point while simultaneously rotating [11].
Table 1. Classification of mobile robots based on the motion system and type of mobility.
point while simultaneously rotating [11].
Table
Table1.1.Classification
Classificationofofmobile
mobilerobots
robotsbased
basedononthe
themotion
motionsystem
systemand
andtype
typeofofmobility.
mobility.
Classification
of
robots
based
motion
system
and type
mobility.
TableTable
1.Table
Classification
of mobile
robots
based
onon
thethe
motion
system
and
type
of
mobility.
1.1.Classification
ofmobile
mobile
robots
based
on
the
motion
system
andof
type
of mobility.
Motion
System
Based
on
Motion
System
Based
on
Table 1. Classification of mobile
robots
based
on
the
motion
system
and
type
of
mobility.
Motion
System
Based
on
Motion
System
Based
Wheels Motion
System
Based
onon on
Motion
System
Based
Legs
Ball
Wheels
Wheels
Wheels
Universal Wheels
Omnidirectional
Ball
Legs
Motion
System Based on
Ball
Legs
Wheels
Ball
Legs
Universal
Omnidirectional
Universal
Omnidirectional
Ball Ball
Legs Legs
Universal
Omnidirectional
Universal
Omnidirectional
Universal Wheels
Omnidirectional
Ball
Legs
Universal
Omnidirectional
(a)
(a)
(a)
Not
omnidirectional
(b)
(c)
(d)
(e)
(b)
(b)
(b) (b)
(b)
(b)
(c)(c) Omnidirectional (d)
(e)
(d)
(e)
(a)
(c) (c)
(d) (d)
(e) (e)
Type of mobility
(a) (a)
(c)Omnidirectional
(e)
Not
Notomnidirectional
omnidirectional
Omnidirectional (d)
(a)
(c) Omnidirectional(d)
(e)
Not omnidirectional
Omnidirectional
Not omnidirectional
Omnidirectional
Type
Typeofofmobility
mobility
Not omnidirectional
Not omnidirectional
Omnidirectional
Type mobile
of mobility
Type
ofrobots
mobility
Currently, in order to achieve mobility
are typically based on wheels. The use of
Type
of mobility
Type
of
mobility
wheels
is
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energy
efficient
than
legged
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treaded
robots
on
hard and
smoot
Currently,
based
on
wheels.
The
Currently,ininorder
ordertotoachieve
achievemobility
mobilitymobile
mobilerobots
robotsare
aretypically
typically
based
onsurfaces
wheels.[12–14].
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Currently,
order
to
achieve
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mobile
robots
are typically
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use
Currently,
in order
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mobile
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The
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mobile
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fixed
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with
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efficient
than
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or
treaded
robots
on
hard
and
smoot
surfaces
[12–14].
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is
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efficient
than
legged
or
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robots
on
hard
and
smoot
surfaces
[12–14].
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inenergy
order
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achieve
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mobile
robots
are These
typically
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use
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order
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mobile
robots
are
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more
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than
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or DOF
treaded
on robots,
hard
and
smoot
surfaces
[12–14].
degrees-of-freedom
(DOF)
instead
of three
(x,y,θ).
likeon
for
example
domestic
wheels
is more
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efficient
than
legged
or
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robots
on
hard
and
smoot
surfaces
The
popular
wheeled
mobile
robots
use
two
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with
two
Themost
most
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independent
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driving
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with
two
is
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energy
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than
legged
or
treaded
robots
on
hard
and
smoot
surfaces
[12–14].
The
most
cleaners,
havewheeled
only wheeled
twoefficient
actuators,
requires
lesstwo
space
to rotate
around
any
point
and
this wheels
also
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is more
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than
legged
or
treaded
robots
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hard
and
smoot
surfaces
[12–14].
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most
popular
mobile
robots
use
independent
fixed
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wheels
with
two
The
most
popular
mobile
robots
use
two
independent
fixed
driving
with
two
degrees-of-freedom
(DOF)
instead
of
three
DOF
(x,y,θ).
These
robots,
like
for
example
domestic
degrees-of-freedom
(DOF)
instead
of
three
DOF
(x,y,θ).
These
robots,
like
for
example
domestic
three
DOF,
but(DOF)
their
limitation
that
they
cannot
perform
holonomic
motion
such
as
sideways
popular
wheeled
mobile
robots
use
two
independent
fixed
driving
wheels
with
two
The
most
popular
wheeled
mobile
robots
use
two
independent
fixed
driving
wheels
with
two
degrees-of-freedom
instead
ofis three
DOF
(x,y,θ).
These
robots,
like
for
example
domestic
degrees-of-freedom
(DOF)
instead
of
three
DOF
(x,y,θ).
These
robots,
like
fordegrees-of-freedom
example
domestic
cleaners,
have
only
two
actuators,
requires
less
space
to
around
any
point
and
this
allows
cleaners,
haveof
only
two
actuators,
requires
less
space
torotate
rotateexample
around
any
pointthis
and
thisalso
also
allows
movements.
An
example
of
this
type
of less
mobile
robot
isrotate
shown
on
Table
1a,
where
configuration
(DOF)
instead
DOF
(x,y,θ).
These
robots,
domestic
have
only
degrees-of-freedom
(DOF)
instead
of
three
DOF
(x,y,θ).
These
robots,
like
forcleaners,
example
domestic
cleaners,
have have
onlythree
two
actuators,
requires
space
tolike
around
any
point
and
this
also
cleaners,
only
two
actuators,
requires
less
space
tofor
rotate
around
any
point
and
this allows
also
allows
three
but
limitation
isisthat
they
threeDOF,
DOF,
buttheir
theirfour
limitation
thatwheels.
theycannot
cannotperform
performholonomic
holonomicmotion
motionsuch
suchasassideways
sideways
is
equipped
with
fixed
universal
cleaners,
have
only
two
actuators,
requires
less
space
to
rotate
around
any
point
and
this
also
allows
two
actuators,
requires
less
space
to
rotate
around
any
point
and
this
also
allows
three
DOF,
but
their
threethree
DOF,DOF,
but their
limitation
is that
cannot
perform
holonomic
motion
such such
as sideways
but their
limitation
is they
that they
cannot
perform
holonomic
motion
as sideways
movements.
ofofmobile
on
ToAn
overcome
thisof
limitation,
other
mobile
robots
an omnidirectional
motion
system,
like for
movements.
Anexample
example
ofthis
thistype
type
mobilerobot
robotisuses
isshown
shown
onTable
Table1a,
1a,where
wherethis
thisconfiguration
configuration
three
DOF,
but
their
is
that
they
cannot
perform
holonomic
motion
asexample
sideways
movements.
An example
oflimitation
this
type
ofholonomic
mobile
robot
is shown
on
1a, where
this such
configuration
limitation
is
that
they
cannot
perform
motion
such
as Table
sideways
movements.
An
of
movements.
An
example
of
this
type
of
mobile
robot
is
shown
on
Table
1a,
where
this
configuration
example,
mobile
robots
equipped
with steerable and coordinated driving wheels and
isisequipped
with
four
universal
wheels.
equipped
with
fourfixed
fixed
universal
wheels.
movements.
An
example
of this
type
ofwheels.
mobile
robotthis
is shown
on Table is
1a,equipped
where this
configuration
is equipped
with
four
fixed
universal
wheels.
this
type
of
mobile
robot
is
shown
on
Table
1a,
where
configuration
with
four
fixed
is equipped
with
four
fixed
universal
omnidirectional
mobile
robotsother
basedmobile
in wheels,
ball-shaped
wheel
robots or legged
robots.
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To
overcome
limitation,
robots
uses
motion
system,
like
To
overcomethis
this
limitation,
other
mobile
robots
usesananomnidirectional
omnidirectional
motion
system,
likefor
for
is
with
four
universal
wheels.
Toequipped
overcome
this
limitation,
otherother
mobile
robots
uses
an
omnidirectional
motion
system,
like for
To
overcome
thisfixed
limitation,
mobile
robots
uses
an omnidirectional
motion
system,
like for
universal
wheels.
devices
offer
interesting
features
when
operating
in
tight
spaces.
Table
1b
shows
a
mobile
robot
example,
example, mobile
mobile robots
robots equipped
equipped with
with steerable
steerable and
and coordinated
coordinated driving
driving wheels
wheels and
and
Tomobile
overcome
this
limitation,
other
mobile
robots
uses
an
omnidirectional
motion
system,
like
for
example,
robots
equipped
with
steerable
andan
coordinated
driving
wheels
and
example,
mobile
robots
equipped
with
steerable
and
coordinated
driving
wheels
design
equipped
with
steerable
wheels
published
by uses
Wada
and
Mori
[15]. They
allow
both
rotation
To
overcome
this
limitation,
other
robots
omnidirectional
motion
system,
likeand
for
omnidirectional
mobile
robots
inmobile
ball-shaped
wheel
omnidirectional
mobile
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based
inwheels,
wheels,
ball-shaped
wheelrobots
robotsororlegged
leggedrobots.
robots.These
These
example,
mobile
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and
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mobile
based
insteerable
wheels,
ball-shaped
wheel
robots
orovercome
legged
robots.
These
and
also sideways
motion,
but
not
simultaneously.
This limitation
canrobots
bewheels
by
using
a These
omnidirectional
mobile
robots
based
in wheels,
ball-shaped
wheel
or and
legged
robots.
example,
mobile
robots
equipped
with
and
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driving
omnidirectional
devices
devicesoffer
offerinteresting
interestingfeatures
featureswhen
whenoperating
operatinginintight
tightspaces.
spaces.Table
Table1b1bshows
showsa amobile
mobilerobot
robot
holonomic
omnidirectional
motion
system,
which
can
inspaces.
all
directions
at
omnidirectional
mobile
robots
based
in wheels,
wheel
robots
orany
legged
robots.
These
devices
offer offer
interesting
features
when
operating
in ball-shaped
tight
spaces.
Table
1b
shows
atime
mobile
robot
devices
interesting
features
when
operating
inmove
tight
Table
1b
shows
awithout
mobile
robot
mobile
robots
based
insteerable
wheels,
ball-shaped
wheel
robots
or
legged
robots.
These
devices
offer
design
equipped
with
wheels
published
by
Wada
and
Mori
[15].
They
allow
both
rotation
design
equipped
with
steerable
wheels
published
by
Wada
and
Mori
[15].
They
allow
both
rotation
changing
wheel
direction,
because
they
can
achieve
3-DOF
motion
on Mori
a[15].
2-dimensional
plane
thenrotation
devices
offer
interesting
features
when
operating
in Wada
tight
spaces.
Table
1bThey
shows
aand
mobile
robot
design
equipped
with
steerable
wheels
published
by
Wada
and
Mori
They
allow
both
rotation
design
equipped
with
steerable
wheels
published
by
and
[15].
allow
both
interesting
features
when
operating
in
tight
spaces.
Table
1b
shows
a
mobile
robot
design
equipped
and
also
sideways
motion,
but
not
simultaneously.
This
limitation
can
be
overcome
by
using
and also
sideways
motion,
but not simultaneously.
This limitation can be overcome by usinga a
thesideways
main
limitation
is steerable
wheel
equipped
with
wheels
published This
by Wada
and
Mori
They
allow
rotation
and design
also
motion,
but slippage
not
simultaneously.
limitation
can [15].
be
overcome
byboth
using
a
and
also
sideways
motion,
but
not[14].
simultaneously.
This
limitation
can
be overcome
by
using
a
with
steerable
wheels
published
by
Wada
and
Mori
[15].
They
both
rotation
and
also
sideways
holonomic
omnidirectional
motion
system,
which
can
inallow
atatany
without
holonomic
omnidirectional
motion
system,
which
can
move
inallalldirections
directions
any
time
without
Holonomic
robots
are
based
in
the
use
of three
ormove
four
omnidirectional
wheels,
seetime
Table
1c,
and
also
sideways
motion,
but
not
simultaneously.
This
limitation
can
be
overcome
by
using
a
holonomic
omnidirectional
motion
system,
which
can
move
in
all
directions
at
any
time
without
holonomic omnidirectional motion system, which can move in all directions at any time without
motion,
butwheel
not
This
limitation
can be
overcome
by
using
a holonomic
omnidirectional
changing
direction,
they
can
3-DOF
motion
on
a a2-dimensional
and
which
aresimultaneously.
composed
by
several
passive
rollers
or
balls
whose
axes
are
tangent toplane
the
wheel
changing
wheel
direction,because
because
they
canachieve
achieve
3-DOF
motion
on
2-dimensional
plane
andthen
then
holonomic
omnidirectional
motion
system,
which
can
move
in aall
directions
atplane
anyplane
time
without
changing
wheel
direction,
because
they can
achieve
3-DOF
motion
on
2-dimensional
and then
changing
wheel
direction,
because
they
can achieve
3-DOF
motion
on
a 2-dimensional
and
then
circumference,
and
free move
to
rotate.
three-wheeled
mobilechanging
robots can wheel
have three
the
limitation
is
slippage
[14].
motion
system,
which
can
in The
all
directions atomnidirectional
any time without
direction,
themain
main
limitation
iswheel
wheel
slippage
[14].
changing
wheel
direction,
because
they[14].
can achieve 3-DOF motion on a 2-dimensional plane and then
the main
limitation
is
wheel
slippage
[14].
theindependent
main
limitation
is
wheel
slippage
actuators
and
they
achieve
independent
translational
one
rotational
DOF,
Holonomic
are
based
incan
use
three
ororfour
wheels,
see
Table
Holonomic
robots
are
based
inthe
the
useof
of2-dimensional
three
fouromnidirectional
omnidirectional
wheels,
seelimitation
Table1c,
1c, is
because
they limitation
can robots
achieve
3-DOF
motion
on
atwo
plane andand
then
the
main
the
main
is wheel
slippage
[14].
Holonomic
robots
are
based
in
theinuse
of
three
or
four
omnidirectional
wheels,
seedomestic
Table
1c, 1c,
Holonomic
robots
are
based
the maneuver
use
of three
or four
omnidirectional
wheels,
see Table
for
the
total
of
3-DOF
on
a
flat
surface
and
and
navigate
in
tight
spaces
such
as
in
which
are
by
which
arecomposed
composed
byseveral
severalpassive
passiverollers
rollersororballs
ballswhose
whoseaxes
axesare
aretangent
tangenttotothe
thewheel
wheel
wheel
slippage
[14]. robots
Holonomic
are
based
inpassive
therollers
use
ofor
three
four
omnidirectional
wheels,
see
Table
1c,
which
are
composed
by several
balls
whose
axes
are tangent
towhen
the
wheel
which
are composed
by
several
rollers
or orballs
axes
are
tangent
to they
the
wheel
environments.
However
due
topassive
their
high
center
of
gravity
theywhose
have
stability
problems
circumference,
and
totorotate.
three-wheeled
omnidirectional
mobile
can
three
circumference,
andfree
free
rotate.The
The
three-wheeled
omnidirectional
mobilerobots
robots
canhave
haveTable
three1c,
Holonomic
robots
are
based
in
the
use
of
three
or
four
omnidirectional
wheels,
see
which
are
composed
by
several
passive
rollers
or
balls
whose
axes
are
tangent
to
the
wheel
circumference,
and
free
to
rotate.
The
three-wheeled
omnidirectional
mobile
robots
can
have
three
are
moving
on
a
ramp
because
of
the
triangular
contact
area
with
the
ground
[5,16].
This
stability
circumference, and free to rotate. The three-wheeled omnidirectional mobile robots can have three
independent
actuators
and
they
achieve
two
translational
and
one
DOF,
independent
actuators
and
theycan
can
achieve
twoindependent
independent
translational
andtangent
onerotational
rotational
DOF,
which
are
composed
by
several
passive
rollers
or
balls
whose
axesmobile
are
to have
the
wheel
problem
canactuators
be
overcome
using
four
wheels
with
4-DOF
[11].
circumference,
and
freethey
to by
rotate.
The
three-wheeled
omnidirectional
robots
can
three
independent
actuators
and
can
achieve
two
independent
translational
and
one
rotational
DOF,
independent
and
they
can
achieve
two
independent
translational
and
one rotational
DOF,
for
the
total
of
3-DOF
on
a
flat
surface
and
maneuver
and
navigate
in
tight
spaces
such
as
in
domestic
for
the
total
of
3-DOF
on
a
flat
surface
and
maneuver
and
navigate
in
tight
spaces
such
as
in
domestic
circumference,
and
free
to
rotate.
The
three-wheeled
omnidirectional
mobile
robots
can
have
three
Table
1d
shows
a
robot
with
ball-shaped
wheels,
such
as
the
design
proposed
by
West
and
independent
actuators
and
they
can
twoand
independent
translational
and
rotational
DOF,
for the
of 3-DOF
on a flat
andachieve
maneuver
navigate
in
tight
spaces
suchone
as in
domestic
fortotal
the total
of
3-DOF
on
asurface
flat
surface
and maneuver
and navigate
in tight
spaces
such
as
in domestic
environments.
However
due
to
high
center
of
they
have
stability
problems
when
environments.
However
due
totheir
their
high[18–20]
center
ofgravity
gravity
they
have
stability
problems
whenthey
they
Asada
[17]
canand
run
any
direction
not
over
rough
grounds
orspaces
steps.
Table
1as
shows
independent
actuators
can
achieve
two
independent
translational
and
one
for
the total
ofthat
3-DOF
on they
ain
flat
surface
and
maneuver
and
navigate
in
tight
suchrotational
inwhen
domestic
environments.
However
due
to
their
high
center
ofbut
gravity
they
have
stability
problems
when
theyDOF,
environments.
However
due
to
their
high
center
of
gravity
they
have
stability
problems
they
are moving
on
because
ofofthe
triangular
contact
area
ground
[5,16].
This
stability
moving
ona aramp
ramp
the
triangular
contact
areawith
withthe
the
ground
[5,16].
stability
that omnidirectional
mobile
robots
based
incenter
wheels
can
have
universal
or
omnidirectional
wheels.
forare
the
total
of 3-DOF
on abecause
flat
surface
and
maneuver
navigate
tight
spaces
suchThis
as
inwhen
domestic
environments.
However
due
to their
high
ofand
gravity
they in
have
stability
problems
they
are moving
on a on
ramp
because
of theoftriangular
contact
area area
with with
the ground
[5,16].
This This
stability
are moving
a ramp
because
the triangular
contact
the ground
[5,16].
stability
problem
problemcan
canbebeovercome
overcomeby
byusing
usingfour
fourwheels
wheelswith
with4-DOF
4-DOF[11].
[11].
environments.
However
due
to
their
high
center
of
gravity
they
have
stability
problems
when
they
are
are
moving
on
a
ramp
because
of
the
triangular
contact
area
with
the
ground
[5,16].
This
stability
problem
can
be
overcome
by
using
four
wheels
with
4-DOF
[11].
problem can be overcome by using four wheels with 4-DOF [11].
Table
Table1d
1dshows
showsa arobot
robotwith
withball-shaped
ball-shapedwheels,
wheels,such
suchasasthe
thedesign
designproposed
proposedby
byWest
Westand
and
problem
can
be
overcome
by
using
four
wheels
with
4-DOF
[11].
moving
onTable
a ramp
the triangular
contact
areasuch
with
ground
[5,16].
This by
stability
problem
Table
1d
shows
a robot
with
ball-shaped
wheels,
asthe
the
design
proposed
West
and and
1d because
shows
aofrobot
with
ball-shaped
wheels,
such
as
the design
proposed
by West
Asada
Asada[17]
[17]that
thatcan
canrun
runininany
anydirection
direction[18–20]
[18–20]but
butnot
notover
overrough
roughgrounds
groundsororsteps.
steps.Table
Table1 1shows
shows
Table
shows
aany
robot
with
ball-shaped
wheels,
asgrounds
the grounds
design
proposed
by
West
and
Asada
[17]
that
can
run
in
[18–20]
but not
over
rough
or steps.
TableTable
1 shows
can
be
overcome
by
using
four
wheels
with
4-DOF
[11].
Asada
[17]1d
that
can
run
indirection
any
direction
[18–20]
but
not such
over
rough
or steps.
1 shows
that
omnidirectional
mobile
robots
based
in
wheels
can
have
universal
or
omnidirectional
wheels.
that omnidirectional mobile robots based in wheels can have universal or omnidirectional wheels.
[17] that can
run
in
anyrobots
direction
but
rough
grounds
or steps. Table
1wheels.
shows
that Asada
omnidirectional
mobile
robots
basedbased
in [18–20]
wheels
cannot
have
universal
or omnidirectional
wheels.
that omnidirectional
mobile
in wheels
canover
have
universal
or omnidirectional
that omnidirectional mobile robots based in wheels can have universal or omnidirectional wheels.
Sensors 2016, 16, 1658
3 of 21
Table 1d shows a robot with ball-shaped wheels, such as the design proposed by West and
Asada [17] that can run in any direction [18–20] but not over rough grounds or steps. Table 1 shows
that omnidirectional mobile robots based in wheels can have universal or omnidirectional wheels.
Alternatively, Table 1e shows also a legged robot because these can move in any direction and can
move on any type of surface, however, the mechanism of legged robots are very complex and have
velocity limitations [21].
In general, the principles of operation of omnidirectional mobile robots are based on kinematic
models. One of the most popular references is the technical report by Muir and Neuman [12] which
formulates the equations of motion of wheel-based mobile robots, incorporating also conventional
omnidirectional and ball wheels. Currently there are a lot authors working in research about the
locomotion of this mobile robots [5,13].
There are many holonomic mobile robot designs available in the literature. The first omnidirectional
mobile robot was proposed in 1987 by Muir and Neuman [22] and was named Uranus. This proposal
was based on introducing a methodology for the kinematic modeling of an omnidirectional wheeled
mobile robot equipped with four omnidirectional wheels which was based in passive rollers arranged
in an overlapping way. These wheels were positioned in pairs on the same axle but with opposite
orientation. Alternatively, in 1996 Wada and Mori [15] proposed a new type of holonomic mobile
robot which was equipped with steerable and coordinated driving wheels using conventional tires to
provide an omnidirectional capability by actuating a wheels axis and a steering axis independently.
There are a several types of omnidirectional wheels but in all of them the principle of function
is based in providing traction in the direction normal to the motor axis, and the use of inner passive
rollers that can slide in the direction of the motor axis. These inner passive wheels, balls or rollers are
placed along the periphery or the main wheels [14,23]. These omnidirectional wheels can be grouped
in four types according to their traces.
Figure 1a shows a wheel design which consists of multiple passive rollers (or inner passive wheels)
whose axes are positioned tangent to the main wheel circumference. This construction cannot avoid
the discontinuous traces and originate an irregular contact with surfaces because of gaps between
successive rollers or wheels, which produce vibrations in the robot. To cancel these effects in these
types of wheels there are some solutions which reduce the size gap between the passive rollers [24].
Mecanum [25] (Figure 1b), was invented in 1973 by Ilon, an engineer working for the Swedish Company
Macanum AB and is other design type of wheel based on rollers arranged in an overlapping way
in such a way that contact between the wheel and the ground is continuous. These wheels are thus
usually positioned in pairs on the same axle but with opposite orientations to form a four-wheel
structure. The drawback of these wheels is the generation of horizontal vibrations because of the
parasite torques which are generated by the fact the contact point moves along a line parallel to the
wheel shaft. The double wheel concept, presented in Figure 1c, is a solution based on two overlapping
parallel wheels. The contact between the assembly wheel and the ground is continuous. This design
generates an important horizontal vibration originated by the gaps between the rotating inner wheels.
Finally, in the design of the Figure 1d, the contact points are in line, which avoids the horizontal
vibrations and the alternated use of passive rollers of different sizes and shapes minimizes the gap
between them thus causing little vertical vibration.
of the parasite torques which are generated by the fact the contact point moves along a line parallel
to the wheel shaft. The double wheel concept, presented in Figure 1c, is a solution based on two
overlapping parallel wheels. The contact between the assembly wheel and the ground is continuous.
This design generates an important horizontal vibration originated by the gaps between the rotating
inner wheels. Finally, in the design of the Figure 1d, the contact points are in line, which avoids the
vibrations and the alternated use of passive rollers of different sizes and shapes minimizes4 of 21
Sensors horizontal
2016, 16, 1658
the gap between them thus causing little vertical vibration.
(a)
(b)
(c)
(d)
Figure 1. Types of omnidirectional wheels and their traces: (a) multiple passive rollers (or inner
Figure 1. Types of omnidirectional wheels and their traces: (a) multiple passive rollers (or inner
passive wheels) whose axes are positioned tangent to the main wheel circumference; (b) with the
passive wheels) whose axes are positioned tangent to the main wheel circumference; (b) with the
rollers arranged in an overlapping way where the contact between the wheels and the ground is
rollerscontinuous;
arranged in
an overlapping way where the contact between the wheels and the ground is
(c) based on two overlapping parallel wheels; (d) based on using alternated passive
continuous;
(c)
based
on two
rollers with different
size overlapping
and shape. parallel wheels; (d) based on using alternated passive rollers
with different size and shape.
Sensors 2016, 16, 1658
4 of 21
3. The Assistant Personal Robot
3. The Assistant Personal Robot
The concept and design of the Assistant Personal Robot (APR, Figure 2) was presented in [9] with
The concept and design of the Assistant Personal Robot (APR, Figure 2) was presented in [9]
the aim to
provide personal assistance services in households or institutions without interfering with
with the aim to provide personal assistance services in households or institutions without interfering
the inhabitants. The APR was designed to be very maneuverable and capable of navigating in tight
with the inhabitants. The APR was designed to be very maneuverable and capable of navigating in
spaces.
physical
design of
the APR
was
inspired
and includes
several
resemblances
with
humans
tightThe
spaces.
The physical
design
of the
APR
was inspired
and includes
several
resemblances
with
in order
to operate,
and moveand
themove
headthe
and
theand
arms
a similar
way. way.
humans
in order maneuver
to operate, maneuver
head
theinarms
in a similar
(a)
(b)
Figure 2. Assistant Personal Robot (APR): (a) CAD design; (b) prototype implementation.
Figure 2. Assistant Personal Robot (APR): (a) CAD design; (b) prototype implementation.
The APR (Figure 2) is a holonomic mobile robot, based on a mechanical structure where all
The APR
a holonomic
robot, based
on a mechanical
structure parts
where all
elements
of (Figure
the APR 2)
areissupported
mademobile
with a combination
of stainless
steel and aluminum
to guarantee
the durability,
resistance
andwith
control
the weight. The
structure steel
of theand
APRaluminum
is divided parts
elements
of the APR
are supported
made
a combination
of stainless
into a circular
base containing
motion
system
based on The
threestructure
omnidirectional
to guarantee
the section
durability,
resistancethe
and
control
the weight.
of thewheels,
APR isand
divided
body,section
which has
two
rotating arms
a multi-touch
panoramic
screenomnidirectional
for interacting with
into aa thin
circular
base
containing
theand
motion
system based
on three
wheels,
humans. This agile, compact and reliable design avoids the presence of sharp edges or projecting
and a thin body, which has two rotating arms and a multi-touch panoramic screen for interacting with
parts, and facilitates its application in a domestic environment. Moreover, the mechanical structure
formed by the body simplifies the application and development of supplementary mechanical
devices.
The APR has a weight of 35 kg with the heavy elements placed on the base and close to the
ground in order to provide a lower center of mass and stable displacement. The APR has a triangular
contact area with the ground because of the three wheeled motion system. The base contains the
Sensors 2016, 16, 1658
5 of 21
humans. This agile, compact and reliable design avoids the presence of sharp edges or projecting parts,
and facilitates its application in a domestic environment. Moreover, the mechanical structure formed
by the body simplifies the application and development of supplementary mechanical devices.
The APR has a weight of 35 kg with the heavy elements placed on the base and close to the ground
in order to provide a lower center of mass and stable displacement. The APR has a triangular contact
area with the ground because of the three wheeled motion system. The base contains the motion
system based on three omnidirectional wheels, the batteries, the LIDAR, and the main electronic
boards. The external design of the base is completed with a bent plastic ABS case, which provides
a flexible protection that will absorb part of the impact in case of collision.
The interactive zone contains a multi-touch panoramic screen and two shoulders with one degree
of freedom in order to move the arms forwards and backwards. The chest and shoulders of the APR are
located
at approximately
1.3 m height which is slightly lower than the shoulders of an average
Sensors
2016, 16, 1658
5 of human.
21
This position was chosen in order to give elderly people direct access to hold the arms of the robot.
average human.
order to give
elderly
people
direct
to hold
The shoulders
of theThis
APRposition
containwas
twochosen
heavyinMicromotor
DC
motors
which
are access
connected
to the
two soft
arms of the robot. The shoulders of the APR contain two heavy Micromotor DC motors which are
arms with a 35 cm separation between them. The arms are 55 cm long just for esthetical reasons and
connected to two soft arms with a 35 cm separation between them. The arms are 55 cm long just for
can be used as a support by elder people when walking or used for basic gesture interactions. The arms
esthetical reasons and can be used as a support by elder people when walking or used for basic
are periodically
moved The
during
forward
displacement
in order
to mimic
the natural
movements
gesture interactions.
armsaare
periodically
moved during
a forward
displacement
in order
to
performed
by
humans
while
walking.
mimic the natural movements performed by humans while walking.
The The
APRAPR
hashas
a height
of of
164
of48
48cm
cmininorder
order
simplify
remote
tele-control
a height
164cm
cmand
andaa width
width of
to to
simplify
the the
remote
tele-control
the mobile
robot
when
passingthrough
throughdoorways,
doorways, small
or or
complicated
paths.
The most
of theofmobile
robot
when
passing
smallcorridors
corridors
complicated
paths.
The most
characteristic
parts
of
the
APR
is
the
holonomic
motion
system
and
this
paper
describes
the
characteristic parts of the APR is the holonomic motion system and this paper describes the mechanical
mechanical
implementation
and
the
control
of
the
motion
system.
implementation and the control of the motion system.
3.1. Mechanical
Implementation
theMotion
MotionSystem
System of
3.1. Mechanical
Implementation
of ofthe
of the
theAPR
APR
The omnidirectional robots can maneuver in small spaces and perform complex trajectory paths.
The omnidirectional robots can maneuver in small spaces and perform complex trajectory paths.
The inclusion of a circular design also minimizes the probability of it getting accidentally caught on
The inclusion of a circular design also minimizes the probability of it getting accidentally caught on
furniture objects such as mats, curtains or clothing. Figure 3 shows the motion system of the APR,
furniture
such
as use
mats,
or clothing. wheels,
Figure shifted
3 shows
theand
motion
system
of the APR,
whichobjects
is based
in the
of curtains
three omnidirectional
120°
composed
of passive
◦
which
is based
the use of three
omnidirectional
wheels, shifted
120motors
and composed
passive
rollers.
rollers.
Thisinthree-wheeled
robot
has three independent
geared DC
attached toofthe
wheels
This three-wheeled
robot3-DOF.
has three
independent
geared
DC motors
attached
to the
and
and they can achieve
Each
wheel has the
same distance,
R, from
its center
to wheels
the center
of they
the can
achieve
3-DOF.
Each wheel has the same distance, R, from its center to the center of the mobile robot.
mobile
robot.
Figure
3. CADmodel
model of
system
based
on theon
usethe
of three
omnidirectional
wheels shifted
Figure
3. CAD
of the
themotion
motion
system
based
use of
three omnidirectional
wheels
120°. ◦
shifted 120 .
In the future, this motion system will include additional suspension in order to minimize the
vertical vibrations caused by the gap between the omnidirectional wheels. The inclusion of a
suspension system based on springs allows the adaptation of the preload to different floor conditions.
The design of the APR is based on the operating principle of a pendulum. This design allows
individual damping and pillar oscillation. The pivoting point is placed over the center of mass,
mainly formed by the batteries, so the forces that appear during the oscillations tend to put the body
to rest, reaching the natural damping equilibrium of the device.
Sensors 2016, 16, 1658
6 of 21
In the future, this motion system will include additional suspension in order to minimize
the vertical vibrations caused by the gap between the omnidirectional wheels. The inclusion of
a suspension system based on springs allows the adaptation of the preload to different floor conditions.
The design of the APR is based on the operating principle of a pendulum. This design allows
individual damping and pillar oscillation. The pivoting point is placed over the center of mass, mainly
formed by the batteries, so the forces that appear during the oscillations tend to put the body to rest,
reaching the natural damping equilibrium of the device.
3.2. Omnidirectional Wheels
The design of the omnidirectional wheels is based on alternating passive rollers with different
Sensors
2016, 16,in
1658
6 ofvertical
21
size and
shapes
order to minimize the gap between the rollers (see Figure 4) which causes
vibration.
This16,implementation
allows wheel spin and perpendicular displacements from 6the
Sensors
2016,
1658
of 21wheel
vibration.
This
implementation allows wheel spin and perpendicular displacements from the wheel
forwarding
direction
and
thus
the
direct
displacement
in
any
direction.
forwarding direction and thus the direct displacement in any direction.
vibration. This implementation allows wheel spin and perpendicular displacements from the wheel
forwarding direction and thus the direct displacement in any direction.
Figure 4. Detail of the gap between the passive rollers.
Figure 4. Detail of the gap between the passive rollers.
Detailinofthe
the gap
the passive
rollers.
The omnidirectionalFigure
wheel4.used
APRbetween
has seven
passive
rollers of two different types,
whose
axes are positioned
tangent
circumference.
Figureof
5atwo
shows
the CAD
design
The
omnidirectional
wheel
usedto
inthe
themain
APRwheel
has seven
passive rollers
different
types,
whose
The omnidirectional wheel used in the APR has seven passive rollers of two different types,
of
the
omnidirectional
wheels
and
Figure
5b
the
prototype
implementation.
Additionally,
Figure
axes are positioned tangent to the main wheel circumference. Figure 5a shows the CAD design 6of the
whose axes are positioned tangent to the main wheel circumference. Figure 5a shows the CAD design
shows a detail
of the passive
rollers
and
shape of the
circumference ofAdditionally,
the wheel.
omnidirectional
wheels
and
Figure
thethe
prototype
implementation.
Figure
6 shows
of the omnidirectional
wheels
and5b
Figure
5b the prototype
implementation. Additionally,
Figure
6
a detail
of
the
passive
rollers
and
the
shape
of
the
circumference
of
the
wheel.
shows a detail of the passive rollers and the shape of the circumference of the wheel.
(a)
(b)
Figure 5. Design of the
(a)omnidirectional wheel: (a) CAD model (b) prototype
(b)implementation.
Figure 5. Design of the omnidirectional wheel: (a) CAD model (b) prototype implementation.
Figure 5. Design of the omnidirectional wheel: (a) CAD model (b) prototype implementation.
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(a)
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(b)
Figure 5. Design of the omnidirectional wheel: (a) CAD model (b) prototype implementation.
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The shape of bracket rollers allows that the small roller can be partially housed in the big roller,
allowing dismiss the gap between the rollers. The circumference of the wheels has an external
diameter of 300 mm and a width of 46 mm. The weight of each omnidirectional wheel implemented
in aluminum is 2.6 kg. The main parts of the omnidirectional wheel (Figure 7) are the roller brackets
Figure 6. CAD section showing the alternate use of the two passive roller types.
(FiguresFigure
8 and 9)
passive
rollers. the
Thealternate
shape of the
rollers
allows
the overlapping
of
6. and
CADthe
section
showing
use bracket
of the two
passive
roller
types.
the rollers in order to reduce the size of the gap (2.5 mm in this implementation) and to minimize the
vertical vibrations in the transitions. Figure 7 shows an exploded view of the basic components of the
Thewheel.
shapeSensors
of bracket
rollers allows that the small roller can be partially housed
in the big roller,
2016, 16, 1658
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allowing dismiss the gap between the rollers. The circumference of the wheels has an external diameter
The shape of bracket rollers allows that the small roller can be partially housed in the big roller,
of 300 mm and aallowing
widthdismiss
of 46 mm.
The weight of each omnidirectional wheel implemented in aluminum
the gap between the rollers. The circumference of the wheels has an external
diameter
of 300 of
mmthe
and aomnidirectional
width of 46 mm. The weight
of each
omnidirectional
wheelroller
implemented
is 2.6 kg. The main
parts
wheel
(Figure
7) are the
brackets (Figures 8
aluminum is 2.6 kg. The main parts of the omnidirectional wheel (Figure 7) are the roller brackets
and 9) and the in
passive
rollers. The shape of the bracket rollers allows the overlapping of the rollers
(Figures 8 and 9) and the passive rollers. The shape of the bracket rollers allows the overlapping of
in order to reduce
the in
size
oftothe
gap
(2.5ofmm
in(2.5
this
and
to minimize
the vertical
the rollers
order
reduce
the size
the gap
mmimplementation)
in this implementation) and
to minimize
the
vertical
vibrations inFigure
the transitions.
Figure
7 shows
an exploded
view
the basic
components
of the of the wheel.
vibrations in the
transitions.
7 shows
an
exploded
view
ofofthe
basic
components
wheel.
Figure 7. Exploded view of the basic components of the wheel.
In the first prototype some parts like rollers and rollers brackets were made in ABS plastic using
the technique of rapid prototyping by fusion deposition modeling (FDM). The main advantage of a
rapid prototyping manufacturing is the simple implementation of any type of complex piece.
Figure
7. Exploded
viewend
of the
basic
components
the wheel.
Nevertheless, several plastic
rollers
brackets
up
breaking
in of
areas
of accumulation of tension due
Figure 7. Exploded view of the basic components of the wheel.
to fatigue stresses.
In the first prototype some parts like rollers and rollers brackets were made in ABS plastic using
the technique of rapid prototyping by fusion deposition modeling (FDM). The main advantage of a
rapid prototyping manufacturing is the simple implementation of any type of complex piece.
Nevertheless, several plastic rollers brackets end up breaking in areas of accumulation of tension due
to fatigue stresses.
Figure 8. Drawing front and side elevation of roller bracket (units in mm).
8. Drawing front and side elevation of roller bracket (units in mm).
FigureFigure
8. Drawing
front and side elevation of roller bracket (units in mm).
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Figure 8 shows the simplified symmetric aluminum implementation of the roller brackets in
order to improve the durability, reduce the cost and simplify the construction of the wheels. The
rollers brackets are simple part which is based on an aluminum plate (8 mm height), cut with a laser
Sensors 2016, 16, 1658
by numeric control and combined with holes also implemented with a computer numeric control to
allow the correct orientation of edges to passive rollers (see Figure 9).
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Figure 8 shows the simplified symmetric aluminum implementation of the roller brackets in
order to improve the durability, reduce the cost and simplify the construction of the wheels. The
rollers brackets are simple part which is based on an aluminum plate (8 mm height), cut with a laser
by numeric control and combined with holes also implemented with a computer numeric control to
allow the correct orientation of edges to passive rollers (see Figure 9).
(a)
(b)
Figure 9. Rollers brackets with axis of big passive roller (green axis) and small passive roller (blue
Figure 9. Rollers
of plastic
big passive
roller
(green axis)
and (b)
small
passive
roller (blue axis).
axis). brackets
(a) Examplewith
madeaxis
of ABS
with a rapid
prototyping
3D printer;
made
of aluminum
with
laser
cut
and
numeric
control.
(a) Example made of ABS plastic with a rapid prototyping 3D printer; (b) made of aluminum with laser
cut and numeric control.
The rollers were also made of aluminum using a numerical control lathe (see Figures 10–13) and
covered with 0.8 mm adherent plastic to increment the grip. The rollers with barrel-shaped are
alternated in order to achieve a continuous contact. The big passive roller has a maximum diameter
In the first
prototype some parts like rollers and rollers brackets were made in ABS plastic
of 44.48 mm and a length of 67.5 mm (see Figures 10 and 11). The small passive roller has a maximum
using the technique
of26.52
rapid
by fusion
(FDM). The main advantage
diameter of
mmprototyping
and a length of 60.5
mm (see deposition
Figures 12 andmodeling
13).
of a rapid prototyping manufacturing is the simple implementation of any type of complex piece.
Nevertheless, several plastic rollers brackets end up breaking in areas of accumulation of tension due
to fatigue stresses.
Figure 8 shows the simplified symmetric aluminum implementation of the roller brackets in order
to improve the durability, reduce the cost and simplify the construction of the wheels. The rollers
(a)
(b)
brackets are simple part which is based on an aluminum plate (8 mm height), cut with a laser by
Figure 9. Rollers brackets with axis of big passive roller (green axis) and small passive roller (blue
numeric control and combined with holes also implemented with a computer numeric control to allow
axis). (a) Example made of ABS plastic with a rapid prototyping 3D printer; (b) made of aluminum
the correct orientation
of edges
passive rollers (see Figure 9).
with laser cut and
numerictocontrol.
The rollers were also made of aluminum using a numerical control lathe (see Figures 10–13)
rollers
made ofplastic
aluminum
using a numerical
control
lathe
(see Figures
10–13) and
and coveredThe
with
0.8were
mmalso
adherent
to increment
the grip.
The
rollers
with barrel-shaped
are
covered with 0.8 mm adherent plastic to increment the grip. The rollers with barrel-shaped are
alternated in order to achieve a continuous contact. The big passive roller has a maximum diameter of
alternated in order to achieve a continuous contact. The big passive roller has a maximum diameter
44.48 mm
and a length of 67.5 mm (see Figures 10 and 11). The small passive roller has a maximum
of 44.48 mm and a length of 67.5 mm (see Figures 10 and 11). The small passive roller has a maximum
diameter
of
26.52
a length
of of
60.5
and
diameter
of mm
26.52 and
mm and
a length
60.5mm
mm(see
(see Figures
Figures 1212and
13).13).
Figure 10. Drawing front and side elevation of big passive roller (units in mm).
Figure 10. Drawing front and side elevation of big passive roller (units in mm).
Figure 10. Drawing front and side elevation of big passive roller (units in mm).
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(a)
(b)
(a)
(b)
Figure 11. Big
passive roller implemented in (a) plastic ABS and (b)
(a)
(b) aluminum.
Figure
11.
Big
passive
roller
implemented
in
(a)
plastic
ABS
and
aluminum.
Figure 11. Big passive roller implemented in (a) plastic ABS and (b)(b)
aluminum.
Figure 11. Big passive roller implemented in (a) plastic ABS and (b) aluminum.
Figure 12. Drawing front and side elevation of small passive roller (units in mm).
Figure 12. Drawing front and side elevation of small passive roller (units in mm).
Figure 12. Drawing front and side elevation of small passive roller (units in mm).
Figure 12. Drawing front and side elevation of small passive roller (units in mm).
(a)
(b)
(a)
(b)
Figure 13. Small
(b) aluminum.
(a) passive roller implemented in (a) plastic ABS and(b)
Figure 13. Small passive roller implemented in (a) plastic ABS and (b) aluminum.
Figure 13. Small passive roller implemented in (a) plastic ABS and (b) aluminum.
Figure 13. Small passive roller implemented in (a) plastic ABS and (b) aluminum.
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3.3. Inverse Kinematic Model
3.3. Inverse Kinematic Model
The functioning principle of the three omnidirectional wheels shifted 120◦ is based on providing
The functioning principle of the three omnidirectional wheels shifted 120° is based on providing
traction in the direction normal to the motor axis while the passive rollers slide in the direction of the
traction in the direction normal to the motor axis while the passive rollers slide in the direction of the
motor axis. The design of the omnidirectional motion allows simultaneous sideways and rotation
motor axis. The design of the omnidirectional motion allows simultaneous sideways and rotation
motion. However, for the shake of simplicity, the study the mobile robot mobility has been divided in
motion. However, for the shake of simplicity, the study the mobile robot mobility has been divided
two parts: translation and rotation. The relationship between the forces exerted by the wheels and the
in two parts: translation and rotation. The relationship between the forces exerted by the wheels and
robot
was based
in a dynamic
model.
The linear
and and
angular
velocities
of the
mobile
robot
the movement
robot movement
was based
in a dynamic
model.
The linear
angular
velocities
of the
mobile
arerobot
the inputs
of
kinematic
model
whereas
the
outputs
are
robot
wheels
velocities.
are the inputs of kinematic model whereas the outputs are robot→wheels velocities.
Figure
the mobile
mobile robot,
robot, v, which
, whichcan
can
represented
Figure1414show
showthe
theinput
inputvelocity
velocity vector
vector of
of the
bebe
represented
in in
polar
form
velocityvector
vectorisisthe
thetarget
target
vector
which
is discomposed
in each
polar
formasas(υ,
(υ,α).
α). This
This velocity
vector
which
is discomposed
in each
wheelwheel
in
in two
twovectors:
vectors:one
one
projection
is
normal
direction
to
the
motor
axis
and
the
other
is
transversal.
projection is normal direction to the motor axis and the other is transversal. The
The
translation
velocities
of wheels
the wheels
b, c been
havenamed
been named
, vtb , and
vtcwheels
. The wheels
translation
velocities
of the
a, b, ca,have
as
, as v, taand
. The
can slidecan
slide
in the
direction
of motor
the motor
thanks
theofuse
of passive
in the
direction
of the
axis axis
thanks
to thetouse
passive
rollers.rollers.
Figure
Kinematic
diagram
translation
mobile
robot:
define
the velocity
Figure
14.14.
Kinematic
diagram
of of
thethe
translation
of of
thethe
mobile
robot:
v andand
α define
the velocity
vector
,
,
and
are
the
velocity
in
each
of
the wheels
vector
of
the
mobile
robot
in
polar
form
and
of the mobile robot in polar form and vta , vtb , and vtc are the velocity in each of the wheels
caused
by translation.
bycaused
translation.
Figure 15 shows the projection on the normal direction,
,
, and
due to the rotation ω.
Figure
15 shows
the projection
on the normal
vra , vrb , and vrc due to the rotation ω.
The
rotational
movement
provides equivalent
speeddirection,
at each wheel.
The rotational
movement
providesthe
equivalent
speed
each wheel.
The equations
that describe
velocity of
each at
wheel,
, in terms of rotation and translation
The
equations
that
describe
the
velocity
of
each
wheel,
ν
,
in
terms of rotation and translation are
w
are Equations (1) and (2):
Equations (1) and (2):
, ,
, ,
,
,
,
(1)
νw (νa , νb , νc ) = vtraslation (νta , cos
νtb , ν30
(1)
tc ) + vrotation ( νra , νrb , νrc ),
∙
cos
150
∙
,
, ,
(2)


cos
cos (270
30 − α)


ν (νa , νb , νof
v ·  cos (150
· R,
(2)
− αmodule
) +ω
c) =
, ,
are thew velocity
the wheels,
is the
of the mobile robot velocity,
where
α)
is the angular orientation expressed in degrees, cosis(270
the −
rotation
of the mobile robot and
is the
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where νw (νa , νb , νc ) are the velocity of the wheels, v is the module of the mobile robot velocity, α is the
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angular orientation expressed in degrees, ω is the rotation of the mobile robot and R is the distance
between
wheelsthe
and
the center
mobile
mm).
Equations
(1) and
(2) can
be used
distancethe
between
wheels
and theofcenter
of robot
mobile(220
robot
(220 mm).
Equations
(1) and
(2) can
be
to used
estimate
the
velocity
of
the
wheels
required
to
move
the
mobile
robot
at
a
desired
velocity
and
to estimate the velocity of the wheels required to move the mobile robot at a desired velocity
angular
rotation.
and angular rotation.
Figure 15. Kinematic diagram of the rotation of the mobile robot:
is the rotation and
,
, and
Figure 15. Kinematic diagram of the rotation of the mobile robot: ω is the rotation and vra , vrb , and vrc
are the velocity in each of the wheels caused by rotation motion.
is the distance between the
are the velocity in each of the wheels caused by rotation motion. R is the distance between the wheels
wheels and the center of mobile robot.
and the center of mobile robot.
3.4. Kinematic Model
3.4. Kinematic Model
The kinematic model of the mobile robot is based on the analysis of the rotation of the wheels in
The
of the
mobile
robot
is based
the mobile
analysis
of the
rotation
of the wheels
order tokinematic
estimate υmodel
(in m/s),
α (in
rad) and
ω (in
rad/s) on
of the
robot.
The
APR estimates
the
in velocity
order toof
estimate
υ
(in
m/s),
α
(in
rad)
and
ω
(in
rad/s)
of
the
mobile
robot.
The
APR
the wheels by using the encoders in the motors of the wheels. The computationestimates
of the
thekinematic
velocity model
of the wheels
by using
encoders
in the motors
of the wheels.
computation
from Equation
(2) the
is not
trivial because
it is a non-linear
system.The
In this
paper, the of
thekinematic
kinematic
model
from
Equation
not trivial
because
it is abecause
non-linear
In this
model
of the
mobile
robot(2)
hasisbeen
obtained
graphically
onlysystem.
the velocities
ofpaper,
the
are available.
16 shows
thehas
kinematic
diagramgraphically
of the motion
of the mobile
robot,
where of
thewheels
kinematic
model ofFigure
the mobile
robot
been obtained
because
only the
velocities
, , are
velocityFigure
of the wheels,
represented
on thediagram
projections
directions
of mobile
the wheels;
the wheels
arethe
available.
16 shows
the kinematic
of of
thethe
motion
of the
robot,
, νa ,, νb , ν, c are
, the
,
are the projections
of therepresented
possible velocity
vector
of motion;
and directions
, ,
arethe
where
velocity
of the wheels,
on the
projections
of the
of
the
points
cuts.
The
interpretation
of
Figure
16
is
as
follows:
the
dashed
lines
show
the
normal
wheels; wa , wb , wc , r a , rb , rc are the projections of the possible velocity vector of motion; and P1 , P2 , P3
to the
motor
of each wheel
the 16
arrow
of rotation
of
aredirection
the points
cuts.
Theaxis
interpretation
of and
Figure
is asindicates
follows:the
thepositive
dasheddirection
lines show
the normal
the wheel
andmotor
on which
project
the and
speed
each wheel,
which
it is a sum
of the speed
of
direction
to the
axis we
of each
wheel
theofarrow
indicates
the positive
direction
of rotation
translation
and
rotation.
In
the
case
of
no
rotation
in
the
displacement
of
the
mobile
robot
(Figure
14),
of the wheel and on which we project the speed of each wheel, which it is a sum of the speed of
the velocity vector of translation are at some point at the normal line established at the end of the
translation
and rotation. In the case of no rotation in the displacement of the mobile robot (Figure 14),
vector speed of each wheel, and then the solution is the cutoff point. In the case of rotation in the
the velocity vector of translation are at some point at the normal line established at the end of the
displacement of the mobile robot (Figure 15), the velocity vector is equidistance to the normal lines
vector speed of each wheel, and then the solution is the cutoff point. In the case of rotation in the
because the rotational movement provides equivalent speeds at each wheel.
displacement of the mobile robot (Figure 15), the velocity vector is equidistance to the normal lines
because the rotational movement provides equivalent speeds at each wheel.
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Figure 16. Graphic diagram to solve the kinematic of mobile robot based on use the projections of the
Figure 16. Graphic diagram to solve the kinematic of mobile robot based on use the projections of the
velocity vectors of each wheel.
velocity vectors of each wheel.
The equations to calculate the projection line of velocity of each wheel
,
, are:
The equations to calculate the projection line of velocity of each wheel r a , rb rc , are:
:
tan 30 ∙
∙ cos 30
√3
√3
√3 1
∙
∙
!
√
√
22
3√
2 2
2 3
3
3
3 1 (3)
r a : y = −tan30 · x + (v a · cos30 + v a · tan30
+
·
2 )√3= − 3 · x + v a
√3· sin30
2
3 2
:
∙
(3)
3
3√
√
2
3
223
· x30
+
v a ∙ tan 30 ∙ sin 30
:
tan 30r a∙ : y = − 3∙ cos
3
√3
√3 √3 1 √
!
√
√
∙
∙
2
2
3
2
3 2 23
3
3 1 (4)
rb : y = tan30 · x + ((−vb ) · cos30 + (−vb ) · tan30 · sin30) =
· x − vb
+
·
3
2
3 2
2 √3
√3
(4)
√
: √
∙
2 3
3
232 3
rb : y =
·x−
v
(5)
3:
3 b
rc : as
x=
vc
This equidistant point can be obtained
the−centroid
of the triangle formed, which can be (5)
obtained using the cutoffs of
,
,
∙ tan 30 ∙ sin 30
:
This equidistant point can be obtained as the centroid of the triangle formed, which can be
obtained using the cutoffs of r a , rb , rc :
√3
⟺
:
,
rc ⇔ rb : P1 =
⟺
:
2
3
!
√
2
3
−vc , −
(vc + 2vb )
3
√3
,
2
3
!
√
2
3
rc ⇔ r a : P2 = −vc ,
(vc + 2v a )
3 √3
⟺ :
,
3
!
√
2
3
r a ⇔ rb : P3 = v a + vb ,
(v a − vb )
2
√3
3
,
,
!
√ 3
3
3
3
v a + vb − 2vc 2 3 (v a − vb )
P1x + P2x + P3x P1y + P2y + P3y
,
=
,
G=
3
3
3
3
(6)
(6)
(7)
(7)
(8)
(8)
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Andthen,
then,the
theequations
equationsfor
forcalculating
calculatingthe
thevelocity
velocityvector
vector of
ofthe
themobile
mobilerobot,
robot,→
whichcan
canbe
be
v ,,which
And
represented
in
polar
form
as
(υ,
α)
and
ω
are:
represented in polar form as (υ, α) and ω are:
v=
q
2
s
( Gx )2 + Gy
2
=
2
2 v a + vb − 2vc 2 (v a − vb )2
+
3
3
3
3
√3
!
√
2
3 (v a − v2b )
v a + vb − 2vc
2
ω∙
v a + vb − 2vc
3
− (−vc )
ω · R = Gx − r c =
3
v a + vb + vc
ωω=
3
3R
α 90 tan
α = 90 − tan−1
(9)
(9)
(10)
(10)
(11)
(11)
being v the
thevelocity
velocityofof
the
mobile
robot,
α the
angular
orientation
ω angular
the angular
rotation
of
being
the
mobile
robot,
α the
angular
orientation
andand
ω the
rotation
of the
the
mobile
robot.
mobile robot.
4.
Control of
of the
the Motion System
4. Control
The
system of
of the
theAPR
APRisiscontrolled
controlled
with
ARM
Cortex-M4
microcontroller
[9].
The motion system
with
an an
ARM
Cortex-M4
microcontroller
[9]. The
The
control
of motion
the motion
is based
oninformation
the information
provided
byencoders
the encoders
themotors
DC motors
control
of the
is based
on the
provided
by the
of theofDC
of the
of
the mobile
The magnetic
encoders
available
in the
DCgenerate
motors generate
three
mobile
robot. robot.
The magnetic
encoders
available
in the low
costlow
DC cost
motors
three impulses
impulses
per
turn
and
the
motor
velocity
is
estimated
by
counting
the
time
between
the
pulses
per turn and the motor velocity is estimated by counting the time between the pulses with a 16with
bits
atimer
16 bitscounter
timer counter
so are
theretree
are velocity
tree velocity
estimates
motor
turn.
Thewheel
wheelvelocity
velocity is directly
so there
estimates
perper
motor
turn.
The
directly
estimated
estimated by
by dividing
dividing the
the motor
motor velocity
velocity by
by the
the fixed
fixed mechanical
mechanical gear
gear ratio
ratio of
of the
the DC
DC motors
motors (43:1).
(43:1).
The
microcontroller
generates
three
Pulse
Width
Modulation
(PWM)
signals
which
are
expressed
The microcontroller generates three Pulse Width Modulation (PWM) signals which are expressedin
in
percentage
H-bridges
in in
order
to convert
thethe
12 V
V atVfull
charge)
of the
percentage(%)
(%)and
andapplied
appliedtotothree
three
H-bridges
order
to convert
12(13.8
V (13.8
at full
charge)
of
battery
into into
an average
DC voltage
in the
Figure
17 shows
the wheel
labeling
used:used:
wheel
1 is
the battery
an average
DC voltage
inmotors.
the motors.
Figure
17 shows
the wheel
labeling
wheel
the
wheel
2 is the
and wheel
3 is the
wheel.
1 isfront-left,
the front-left,
wheel
2 isfront-right,
the front-right,
and wheel
3 isback
the back
wheel.
Figure17.
17.Detail
Detailof
ofthe
thelabeling
labelingand
andpositive
positiveangular
angularvelocity
velocityof
ofthe
thewheels.
wheels. The
Thegreen
greenarrow
arrowdepicts
depicts
Figure
the
front
of
the
mobile
robot.
the front of the mobile robot.
The DC motors are powerful enough to displace the 35 kg of the mobile robot with a forward
The DC motors are powerful enough to displace the 35 kg of the mobile robot with a forward
velocity comparable with that of a walking human. Figure 18 shows the profile of the velocity of the
velocity comparable with that of a walking human. Figure 18 shows the profile of the velocity of the
wheels of the APR in case of applying a fixed PWM of 62%, 40%, 25% during 4 s to the motors of the
wheels of the APR in case of applying a fixed PWM of 62%, 40%, 25% during 4 s to the motors of the
APR in an open loop operation without any feedback control. The average velocities of the wheels
APR in an open loop operation without any feedback control. The average velocities of the wheels
were approximately 45, 30 and 15 rpm respectively. Figure 19 shows an image of the initial and
were approximately 45, 30 and 15 rpm respectively. Figure 19 shows an image of the initial and ending
ending mobile robot position obtained after this experiment. Figure 18 shows that, in this case, the
mobile robot position obtained after this experiment. Figure 18 shows that, in this case, the wheels
wheels reach the 80% of the maximum speed in only 0.1 s which means that the mobile robot reaction
is very fast but then the mechanical stress suffered by the onboard mechanical and electronic elements
Sensors 2016, 16, 1658
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reach the 80% of the maximum speed in only 0.1 s which means that the mobile robot reaction is
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2016,
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very fast
but
then
the mechanical stress suffered by the onboard mechanical and electronic14elements
is also very high. Figure 18 also shows that there are an unwanted peak prior velocity stabilization.
is also very high. Figure 18 also shows that there are an unwanted peak prior velocity stabilization.
Despite
thesethese
citedcited
effects,
thethe
trajectory
ofof
the
anopen
openloop
loopininthe
the
motors is
Despite
effects,
trajectory
theAPR
APRininthe
thecase
case of
of using
using an
motors
very stable
and
visually
predictable.
The
only
drawback
is
a
violent
acceleration
and
deceleration
is very stable and visually predictable. The only drawback is a violent acceleration and deceleration due
to thedue
power
ofpower
the motors
used inused
the in
motion
system.
to the
of the motors
the motion
system.
50
45
40
Velocity (rpm)
35
30
25
20
15
10
5
0
0
Wheel 1
Wheel 2
Wheel 3
1
2
Time (s)
3
4
5
Figure 18. Wheel velocity profile during an open loop operation.
Figure
18. Wheel velocity profile during an open loop operation.
The control of the velocity of the wheels of the mobile robot has been performed by applying a
The
control of
the velocity
of theand
wheels
of the(PID)
mobile
robot to
has
performed
applying
conventional
Proportional,
Integral,
Derivative
controller
thebeen
velocity
of the DCby
motors
in a closedProportional,
loop control. In
general,and
the PID
controller
is designed
to provide
fast reaction
target
a conventional
Integral,
Derivative
(PID)
controller
to the velocity
of thetoDC
motors
changes.
However,
of the
PID alsoisallows
a smoothed
supervised
evolution
in a closed
loop
control.anInadequate
general,design
the PID
controller
designed
to provide
fast reaction
toof
target
the
power
applied
to
the
device
controlled.
In
this
paper,
the
tuning
of
the
PID
controller
constants
changes. However, an adequate design of the PID also allows a smoothed supervised evolution of
has been
performed
a trial controlled.
and error procedure
with the
of obtaining
a smooth,
stableconstants
and
the power
applied
to thebydevice
In this paper,
theaim
tuning
of the PID
controller
visually predictable mobile robot motion. At the end of this subjective manual tuning procedure the
has been performed by a trial and error procedure with the aim of obtaining a smooth, stable and
PID controller has been simplified as a Proportional and Integral (PI) controller with KP and KI values
visually predictable mobile robot motion. At the end of this subjective manual tuning procedure
of 0.01 and 1.50 respectively (Figure 20). As a summary, the control procedure applied is as follows:
the PID
has been
simplified
a aProportional
Integral
controller
KP and KI
(1) acontroller
target velocity
(in rpm)
is definedasfor
wheel; (2) oneand
internal
timer(PI)
is used
to countwith
the encoder
values
of
0.01
and
1.50
respectively
(Figure
20).
As
a
summary,
the
control
procedure
applied
pulses and to estimate the velocity of the DC motor velocity (in rpm); (3) the motor velocity is then is as
follows:
(1) a target
velocity
(in(inrpm)
forthe
a gear
wheel;
(2)(4)
one
timer
is usedwheel
to count
converted
to wheel
velocity
rpm)isbydefined
applying
ratio;
theinternal
target and
measured
velocity pulses
are thenand
compared;
and the
(5) the
difference
is DC
multiplied
KD and
and integrated
the encoder
to estimate
velocity
of the
motor by
velocity
(inKIrpm);
(3) the motor
(cumulated).
In this paper,
the cumulative
converted
in target
PWM and
velocity
is then converted
to wheel
velocity (indifference
rpm) by expressed
applying in
therpm
gearis ratio;
(4) the
percentage
by
applying
a
conversion
factor
of
approximately
1.42%/rpm.
measured wheel velocity are then compared; and (5) the difference is multiplied by KD and KI and
21 showsInthe
profile
of the
of thedifference
wheels of expressed
the APR inin
case
of activating
the in
DCPWM
integratedFigure
(cumulated).
this
paper,
the velocity
cumulative
rpm
is converted
motors during 4 s and controlling the velocity with the proposed closed loop. Figure 21 shows that,
percentage by applying a conversion factor of approximately 1.42%/rpm.
in this case, the wheels reach the 80% of the maximum speed in more than 1 s. The drawback of this
Figure 21 shows the profile of the velocity of the wheels of the APR in case of activating the DC
closed loop control is that the integral controller does not stop the mobile robot instantaneously
motors
during
s and controlling
the
velocity
with
the proposed
loop. Figure
21 shows
because
the4deceleration
profile is
similar
to the
acceleration
profile.closed
In the practice,
the selection
of a that,
in this
case,wheel
the wheels
than 1 creates
s. The adrawback
target
velocityreach
up tothe
the80%
95%ofofthe
themaximum
maximum speed
velocityinofmore
the motors
smooth of
this closed
loop control
is that the
integral
controller
not stop
the mobile
robot
acceleration
and deceleration
profile
that not
saturatedoes
the motors,
reduce
vibrations
and instantaneously
improve the
external
predictability of
the trajectory
of the
robot. In anyprofile.
case, theIn
high
control
system
because
the deceleration
profile
is similar
to mobile
the acceleration
theorder
practice,
the
selection
of the wheel
APR can
force an
the mobile
robot by
the integral
of a target
velocity
up instantaneous
to the 95% ofstop
the of
maximum
velocity
of disconnecting
the motors creates
a smooth
controller
and
short-cuttingprofile
the H-bridges
order to block
the motors
(applying
an electrical
brake). the
acceleration
and
deceleration
that notinsaturate
the motors,
reduce
vibrations
and improve
external predictability of the trajectory of the mobile robot. In any case, the high order control system
of the APR can force an instantaneous stop of the mobile robot by disconnecting the integral controller
and short-cutting the H-bridges in order to block the motors (applying an electrical brake).
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Figure 19. Composite image created to show the relative displacement of the mobile robot originated
Figure
19.
Composite
created
to
show
therelative
relativedisplacement
displacement
the
mobile
robot
originated
Figure
19.
Compositeimage
image
created
to18.
show
the
relative
ofof
the
mobile
robot
originated
Figure
19.
image
to
show
the
of
the
mobile
robot
originated
byComposite
wheel velocity
profilecreated
of Figure
The APR
maintainsdisplacement
the absolute angular
orientation.
byby
wheel
18. The
TheAPR
APRmaintains
maintainsthe
theabsolute
absolute
angular
orientation.
by
wheelvelocity
velocityprofile
profileof
ofFigure
Figure 18.
APR
maintains
the
absolute
angular
orientation.
wheel
velocity
profile
of
Figure
angular
orientation.
Target velocity
Target
Targetvelocity
velocity
Error
Error
Error
KP
KP
KP
KI
RPM  PWM
MOTOR
RPM
 PWM
RPM
 PWM
MOTOR
MOTOR
KI
KI
RPM  PULSE
Figure 20. Detail of the PI controller implementation.
RPM

PULSE
RPM
PULSE
50
Figure
Figure
20.
Detail
ofthe
thePI
PIcontroller
controllerimplementation.
implementation.
Figure20.
20.Detail
Detailof
controller
implementation.
45
Velocity (rpm)
Velocity
(rpm)
Velocity
(rpm)
50
40
50
35
45
45
30
40
40
25
35
35
20
30
15
30
25
10
Wheel 1
25
Wheel 2
20
5
Wheel 3
20
0
15
0
1
2
3
4
5
15
Time (s)
10
Wheel 1
during a closed loop operation.
10 Figure 21. Wheel velocity profile
Wheel
Wheel2 1
5
Wheel
Wheel3 2
5
0
Wheel 3
0
1
2
3
4
5
0
Time (s)
0
1
2
3
4
5
Time (s)
Figure 21. Wheel velocity profile during a closed loop operation.
Figure
duringaaclosed
closedloop
loopoperation.
operation.
Figure21.
21.Wheel
Wheelvelocity
velocity profile
profile during
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Finally,
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the
16 ofwheels,
21
22–24 show the target velocity setpoint applied to the PID of
Finally, Figures 22–24 show the target velocity setpoint applied to the PID of the wheels, the
the evolution
of
the
wheel
estimated
with setpoint
the encoders,
evolution
of the
Finally, Figures 22–24velocity
show the
target velocity
applied and
to thethe
PID
of the wheels,
thePWM
evolution
of the
wheel
velocity
estimated
with
the encoders,
and
the evolution
ofof
thethe
PWM
applied
Finally,
Figures
22–24
show
the
target
velocity
setpoint
applied
to
the
PID
wheels,
the
evolution
of the
wheel velocity
estimated
with
the encoders,
and the evolution
theexample,
PWM applied
applied
to the DC
motors.
This control
system
is simple
but extremely
effective.ofFor
when the
to the DC of
motors.
This control system
is simple
butencoders,
extremelyand
effective.
For example,
when
the
APR
the
wheel
estimated
with the
the predictable
evolution
of trajectories
the when
PWMthe
applied
to
the
DC motors.
This velocity
control
system
is simple
but
extremelyalways
effective.
For example,
APR are
APR evolution
is
remotely
tele-operated
this
control
system
generates
which
is remotely
tele-operated
this control
system
generates
always predictable
trajectories
which
are very
to
the DC motors.
This control
systemsystem
is simple
but extremely
effective. For
example, which
when the
remotely
tele-operated
this control
generates
always
predictable
trajectories
are APR
verytakes
very is
easy
tosupervise.
supervise.
The
combination
of inverse
the
inverse
kinematic
model
and
the control
system
easy
to
The
combination
of
the
kinematic
model
and
the
control
system
takes
full
is
remotely
tele-operated
this controlofsystem
generates
alwaysmodel
predictable
trajectories
whichtakes
are very
easy
to supervise.
The combination
the inverse
kinematic
and the
control system
full
full advantage
of the
theholonomic
holonomic
motion
of the
mobile
robot
inofcase
of severe
trajectory
advantage of
motion
of the
mobile
robot
eveneven
in case
severe
trajectory
changes.changes.
easy
to supervise.
The combination
kinematic
the control
system
takes full
advantage
of the holonomic
motion of
of the
the inverse
mobile robot
evenmodel
in caseand
of severe
trajectory
changes.
advantage of the holonomic motion of the mobile robot even in case of severe trajectory changes.
40
40
40
Velocity
(rpm)
Velocity
(rpm)
Velocity
(rpm)
100
100
100
Velocity
Velocity
Setpoint
Setpoint
Velocity
PWM (%)
PWM (%)
Setpoint
PWM (%)
80
80
80
30
30
30
60
60
60
20
20
20
40
40
40
10
10
10
20
20
20
0
00
0
0
0
1
1
1
2
2
2
3
4
5
3 (s)
4
5
Time
Time
(s)
3
4
5
velocity Time
and (s)
applied motor PWM
PWM
(%)(%)
PWM
PWM
(%)
50
50
50
0
60
6
0
6
Figure 22. Wheel 1. Evolution of wheel
for a target velocity of 44 rpm.
Figure
22.22.
Wheel
1. 1.Evolution
andapplied
appliedmotor
motorPWM
PWM
a target
velocity
ofrpm.
44 rpm.
Figure
Wheel
Evolutionof
ofwheel
wheel velocity
velocity and
forfor
a target
velocity
of 44
Figure 22. Wheel 1. Evolution of wheel velocity and applied motor PWM for a target velocity of 44 rpm.
40
40
40
Velocity
(rpm)
Velocity
(rpm)
Velocity
(rpm)
100
100
100
Velocity
Velocity
Setpoint
Setpoint
Velocity
PWM (%)
PWM (%)
Setpoint
PWM (%)
80
80
80
30
30
30
60
60
60
20
20
20
40
40
40
10
10
10
20
20
20
0
00
0
0
0
1
1
1
2
2
2
3
4
3 (s)
4
Time
Time
3 (s)
4
Time
and (s)
applied motor
5
5
5
PWM
(%)(%)
PWM
PWM
(%)
50
50
50
0
60
6
0
6
Figure 23. Wheel 2. Evolution of wheel velocity
PWM for a target velocity of 29 rpm.
Figure 23. Wheel 2. Evolution of wheel velocity and applied motor PWM for a target velocity of 29 rpm.
Figure
23.23.
Wheel
2. 2.Evolution
andapplied
appliedmotor
motorPWM
PWM
a target
velocity
ofrpm.
29 rpm.
Figure
Wheel
Evolutionof
ofwheel
wheel velocity
velocity and
forfor
a target
velocity
of 29
Velocity
Velocity
Setpoint
Setpoint
Velocity
PWM (%)
PWM (%)
Setpoint
PWM (%)
Velocity
(rpm)
Velocity
(rpm)
Velocity
(rpm)
40
40
40
100
100
100
80
80
80
30
30
30
60
60
60
20
20
20
40
40
40
10
10
10
20
20
20
0
00
0
0
0
1
1
1
2
2
2
3
4
3 (s)
4
Time
Time
3 (s)
4
Time applied
(s)
and
motor
5
5
5
PWM
(%)(%)
PWM
PWM
(%)
50
50
50
0
60
6
0
6
Figure 24. Wheel 3. Evolution of wheel velocity
PWM for a target velocity of 15 rpm.
Figure 24. Wheel 3. Evolution of wheel velocity and applied motor PWM for a target velocity of 15 rpm.
Figure 24. Wheel 3. Evolution of wheel velocity and applied motor PWM for a target velocity of 15 rpm.
Figure 24. Wheel 3. Evolution of wheel velocity and applied motor PWM for a target velocity of 15 rpm.
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17
5. Validation
5. Validation
The validation of the three-wheels holonomic motion system consisted of the comparison of
the trajectory
of the of
mobile
robot obtained
by usingmotion
two alternative
methods:of
(1)the
by comparison
using a SLAM
The validation
the three-wheels
holonomic
system consisted
of
procedure
(used
in [26])
forrobot
precise
absolute
based on the
information
by the
the
trajectory
of the
mobile
obtained
bypositioning
using two alternative
methods:
(1) byprovided
using a SLAM
onboard LIDAR;
usingabsolute
the information
ofbased
the encoders
and the kinematic
model
procedure
(used inand
[26])(2)
forby
precise
positioning
on the information
provided by
the
presented
in this
paper.
objective of
validation
is tomodel
compare
both
onboard
LIDAR;
and
(2) by The
usingmain
the information
of the encoders
andprocedure
the kinematic
presented
measurements.
in
this paper. The main objective of the validation procedure is to compare both measurements.
summarizes eight trajectories of the mobile robot obtained in a first
first validation
validation
Figure 25 summarizes
maintain its orientation
orientation while moving
moving aa straight
straight
experiments. In this case, the mobile robot has to maintain
α (see
(seeFigure
Figure 19
19 for
foraareference).
reference). This
This capability
capability of
of
fixed distance in a predefined angular direction, α
moving without
withoutchanging
changingthe
theorientation
orientation
a characteristic
feature
a holonomic
motion
system
moving
is is
a characteristic
feature
of aofholonomic
motion
system
that
◦ , 90
◦,
that
is validated
in experiment.
this experiment.
The angular
direction
tested
the displacements
(0°,
45°,
is
validated
in this
The angular
direction
tested
in theindisplacements
werewere
(0◦ , 45
90°,◦ ,135°,
the angular
velocity
always
135
180◦ ,180°,
225◦225°,
, 270◦270°,
, 315◦315°)
) and and
the angular
velocity
was was
always
0◦ . 0°.
The procedure
procedureofofeach
eachmeasurement
measurement
follows:
a specific
mobile
robot
velocity,
distance
The
is is
asas
follows:
(1)(1)
a specific
mobile
robot
velocity,
distance
and
and angular
direction
the displacement
is fixed;
the mobile
computes
the angular
velocity
angular
direction
of theofdisplacement
is fixed;
(2) the(2)
mobile
robot robot
computes
the angular
velocity
of the
of the wheels
according
the kinematic
inverse kinematic
model,
thiswill
velocity
will
remainduring
constant
the
wheels
according
the inverse
model, this
velocity
remain
constant
theduring
validation
validation experiment;
(3) robot
the mobile
robot
ideal-time
required
for the according
displacement
experiment;
(3) the mobile
estimates
theestimates
ideal-timethe
required
for the
displacement
the
according model;
the kinematic
model; time
(4) the
internal time
is reset;velocities
(5) the angular
velocities
kinematic
(4) the internal
measurement
is measurement
reset; (5) the angular
of the wheels
are
of the wheels
are stablished
setpoints
of the three
PI controllers
of the
robotisand
the motion
stablished
as setpoints
of theasthree
PI controllers
of the
mobile robot
andmobile
the motion
automatically
is automatically
when
reaching
the experiment,
ideal-time ofthe
thesetpoints
experiment,
setpoints
of the
started;
(6) whenstarted;
reaching(6)the
ideal-time
of the
of thethe
angular
velocities
angular
velocities
of
the
wheels
are
set
to
zero
and
the
PI
controllers
deaccelerate
the
mobile
robot
of the wheels are set to zero and the PI controllers deaccelerate the mobile robot until completely
until completely
stopping
device. Figure
summarizes
the mobile
robot with
trajectories
obtained
stopping
the device.
Figure the
25 summarizes
the 25
mobile
robot trajectories
obtained
the information
with
information
the encoders
and
the
kinematic
model
and with
SLAM
procedure:
the
of
thethe
encoders
and theofkinematic
model
and
with
the SLAM
procedure:
thethe
circle
depicts
the position
circle depicts
position
according
thecross
SLAM
procedure
the cross of
according
the information
of
according
the the
SLAM
procedure
and the
according
theand
information
the encoders;
in both cases
encoders;
in depicts
both cases
small
line also depicts
the final
orientation
the mobile robot
is
athe
small
line also
theafinal
orientation
of the mobile
robot
which isofapproximately
the which
same in
approximately
the
same
in
all
trajectories.
all trajectories.
1200
1000
800
Y position (mm)
600
400
200
0
-200
-400
-600
-800
-1000
-1000
-500
0
X position (mm)
500
1000
Figure 25.
25. Trajectories
followed by
by the
the mobile
mobile robot
robot in
in eight
eight displacements.
displacements. The
The circle
circle depicts
depicts the
the final
final
Figure
Trajectories followed
position
and
orientation
of
the
mobile
robot
obtained
with
the
SLAM
procedure.
The
cross
depicts
position and orientation of the mobile robot obtained with the SLAM procedure. The cross depicts the
the final
position
orientation
of mobile
the mobile
robot
according
the information
of encoders
the encoders
final
position
and and
orientation
of the
robot
according
the information
of the
and and
the
the
kinematic
model.
kinematic model.
The results shown in Figure 25 show small differences in the mobile robot trajectory estimated
The results shown in Figure 25 show small differences in the mobile robot trajectory estimated
with the SLAM procedure and with the encoders and kinematic model of the motion system.
with the SLAM procedure and with the encoders and kinematic model of the motion system. Figure 26
Figure 26 details the absolute error in the position estimated with the LIDAR and the kinematic
details the absolute error in the position estimated with the LIDAR and the kinematic model compared
model compared with the planned displacement. The maximum difference between the desired
with the planned displacement. The maximum difference between the desired theoretical and estimated
theoretical and estimated distance was always lower than 60 mm in all measurements
distance was always lower than 60 mm in all measurements corresponding to a planned straight
corresponding to a planned straight trajectory displacement of 1000 mm with the mobile robot.
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Moreover,displacement
the difference
difference
of
themm
position
estimated
with
the
SLAM procedure
procedure
and of
the
kinematic
trajectory
ofof
1000
with the
mobile with
robot.the
Moreover,
the difference
the
position
Moreover,
the
the
position
estimated
SLAM
and
the
kinematic
model
were
always
lower
than
30
mm,
which
may
be
considered
a
very
low
difference.
Finally,
estimated
with
the
SLAM
procedure
and
the
kinematic
model
were
always
lower
than
30 mm,
model were always lower than 30 mm, which may be considered a very low difference. Finally,
Figure
27
shows
the
differences
between
the
planned
angular
orientation
of
the
mobile
robot
and
which
be considered
a verybetween
low difference.
Finally,
Figure
27 shows of
thethe
differences
between
Figure may
27 shows
the differences
the planned
angular
orientation
mobile robot
and
the values
values estimated
estimated
according the
the
SLAM
procedure
and the
according
the
information
of the
thethe
encoders
the
planned
angular according
orientation
of SLAM
the mobile
robot and
values the
estimated
according
SLAM
procedure
and
according
information
of
encoders
and
the
kinematic
model
which
are
always
lower
than
5°
in
all
trajectories
measured
in
this
first
procedure
and
according
the
information
of
the
encoders
and
the
kinematic
model
which
are
always
and the kinematic model which are always lower than 5° in all trajectories measured in this first
◦
validation
experiment.
The
displacement
estimated
with
the
information
of
the
encoders
has
been
lower
thanexperiment.
5 in all trajectories
measuredestimated
in this first
experiment.
The displacement
validation
The displacement
withvalidation
the information
of the encoders
has been
also
compared
with
manual
measurements
and
with
measurements
obtained
with
an
external
estimated
with
the
information
of
the
encoders
has
been
also
compared
with
manual
measurements
also compared with manual measurements and with measurements obtained with an external
LIDAR
[27]
obtaining
similar
results
than
the
displacement
estimated
with
the
SLAM
procedure.
and
with
measurements
obtained
with
an the
external
LIDAR [27]
obtaining
results
than the
LIDAR
[27]
obtaining similar
results
than
displacement
estimated
withsimilar
the SLAM
procedure.
However, the
theestimated
use of
of the
thewith
SLAM
onboard
procedure
has the
the the
advantage
ofSLAM
an easy
easy
synchronization
displacement
the SLAM
procedure.
However,
use of the
onboard
procedure
However,
use
SLAM
onboard
procedure
has
advantage
of
an
synchronization
between
the trajectories
trajectories
estimated
with both
both between
methods.the trajectories estimated with both methods.
has
the advantage
of an easy
synchronization
between
the
estimated
with
methods.
10
10
00
SLAM estimate
estimate
SLAM
Encoders estimate
estimate
Encoders
Error
Error(mm)
(mm)
-10
-10
-20
-20
-30
-30
-40
-40
-50
-50
11
22
33
44
55
Experiment
Experiment
66
77
88
Figure 26.
26. Absolute
Absolute displacement
displacement error
error obtained
obtained when
when comparing
comparing the
the planned
planned displacement
displacement and
and the
the
Figure
displacement
final
displacement
estimated
with
the
SLAM
procedure
and
with
the
encoders
and
the
kinematic
model.
final displacement estimated with the SLAM procedure and with the encoders and the kinematic model.
66
Relative
Relativeerror
error(Degrees)
(Degrees)
55
44
33
22
11
00
-1
-1
-2
-2
-3
-3
-4
-4
11
22
33
44
55
Experiment
Experiment
SLAM estimate
estimate
SLAM
Encoders
estimate
Encoders estimate
6
7
6
7
88
Figure 27.
27. Absolute
Absolute final
final angular
angular orientation
orientation error
error obtained
obtained when
when comparing
comparing the
the planned
planned angular
angular
Figure
Figure
27. Absolute
final angular
orientation error
obtained when
comparing the
planned angular
orientation and the final orientation estimated with the SLAM procedure and with the encoders and
orientation and the final orientation estimated with the SLAM procedure and with the encoders and
the kinematic model.
the kinematic model.
Finally, Figure
Figure 28 summarizes
summarizes the mobile
mobile robot trajectories
trajectories obtained
obtained with
with the
the information
information of
of the
the
Finally,
Finally,
Figure 28
28 summarizesthe
the mobilerobot
robot trajectories
obtained with
the information
of
encoders and
and the
the kinematic
kinematic models
models and
and with
with the
the SLAM
SLAM procedure.
procedure. In
In this
this second
second validation
validation
encoders
the encoders and the kinematic models and with the SLAM procedure. In this second validation
experiment, the
the mobile robot
robot has to
to maintain its
its orientation while
while moving aa fixed
fixed distance following
following a
experiment,
experiment,
the mobile
mobile robot has
has tomaintain
maintain itsorientation
orientation whilemoving
moving a fixeddistance
distance followinga
curved path.
path. The
The angular
angular direction
direction tested
tested in
in the
the displacements
displacements were
were (0°,
(0°, 45°,
45°,
90°,
135°, 180°,
180°,
225°,
270°,
90°,
225°,
270°,
◦ , 45
◦ , 135°,
◦ , 180
◦ , 225
◦,
acurved
curved path.
The angular
direction tested
in the displacements
were (0
90◦ , 135
315°)
and
the
angular
velocity
was
always
5%
of
the
maximum
velocity.
In
these
measurements,
the
315°)
and
the
angular
velocity
was
always
5%
of
the
maximum
velocity.
In
these
measurements,
the
◦
◦
270 , 315 ) and the angular velocity was always 5% of the maximum velocity. In these measurements,
difference between
between the
the position
position estimated
estimated with
with the
the SLAM
SLAM procedure
procedure and
and with
with the
the encoders
encoders and
and the
the
difference
Sensors 2016, 16, 1658
19 of 21
Sensors 2016, 16, 1658
19 of 21
the difference between the position estimated with the SLAM procedure and with the encoders and the
kinematic
model
was
of1000
1000mm
mmwhereas
whereasthe
thedifference
difference
kinematic
model
wasagain
againlower
lowerthan
than30
30mm
mm in
in aa total
total distance
distance of
◦
between
thethe
estimates
ofthe
themobile
mobilerobot
robot
was
also
lower
than
between
estimatesofofthe
theangular
angular orientation
orientation of
was
also
lower
than
5°. 5 .
1200
1000
800
Y position (mm)
600
400
200
0
-200
-400
-600
-800
-1000
-1000
-500
0
X position (mm)
500
1000
Figure
Trajectories
followedby
bythe
themobile
mobilerobot
robot in
in eight
thethe
final
Figure
28.28.
Trajectories
followed
eight displacements.
displacements.The
Thecircle
circledepicts
depicts
final
position and orientation of the mobile robot obtained with the SLAM procedure. The cross depicts
position and orientation of the mobile robot obtained with the SLAM procedure. The cross depicts the
the final position and orientation of the mobile robot according the information of the encoders and
final position and orientation of the mobile robot according the information of the encoders and the
the kinematic model.
kinematic model.
The comparative empirical results obtained validate the design of the three-wheel holonomic
The
empirical
results obtained
design
of the
three-wheel
holonomic
motioncomparative
system for mobile
robot displacement
and validate
the utilitythe
of the
proposed
inverse
kinematic
model
motion
system
for
mobile
robot
displacement
and
the
utility
of
the
proposed
inverse
kinematic
in order to estimate the relative displacement of the mobile robot according the information ofmodel
the
in order
to estimate
the relative displacement of the mobile robot according the information of the
encoders
of the wheels.
encoders of the wheels.
6. Conclusions
6. Conclusions
This paper presents the design and implementation of the mechanical design of the three-wheel
holonomic
motion
system
implemented
in the Assistant
Personal
Robot design
(APR),ofa the
mobile
robot
This paper
presents
the design
and implementation
of the
mechanical
three-wheel
designedmotion
to operate
at home.
The paper
analyzes
the Personal
inverse and
direct
kinematics
the motion,
holonomic
system
implemented
in the
Assistant
Robot
(APR),
a mobileofrobot
designed
describesatthe
control
system
shows the result
of and
different
experiments
to validate
the
to operate
home.
The
paperand
analyzes
inverse
direct
kinematicsproposed
of the motion,
describes
complete
motion
system.
The
trajectory
of
the
mobile
robot
has
been
estimated
by
using
the
the control system and shows the result of different experiments proposed to validate the complete
information
the trajectory
encoders ofofthe
and
the proposed
This
trajectory
has been
motion
system.ofThe
thewheels
mobile
robot
has been kinematic
estimatedmodel.
by using
the
information
of the
compared
with
the
trajectory
obtained
with
a
SLAM
procedure
based
on
the
information
obtained
encoders of the wheels and the proposed kinematic model. This trajectory has been compared with
an onboard
LIDAR.
Results
have procedure
shown a discrepancy
in both
estimatesobtained
of less than
30 mm
in
theby
trajectory
obtained
with
a SLAM
based on the
information
by an
onboard
distance, and less than 5° in angular orientation for absolute displacements of up to 1000 mm. These
LIDAR. Results have shown a discrepancy in both estimates of less than 30 mm in distance, and less
results confirm the utility of the three-wheel holonomic motion system proposed for the
than 5◦ in angular orientation for absolute displacements of up to 1000 mm. These results confirm the
implementation of an Assistant Personal Robot.
utility of the three-wheel holonomic motion system proposed for the implementation of an Assistant
Personal
Robot.
Acknowledgments:
This work was partially funded by Indra, the University of Lleida, the RecerCaixa 2013
grant, and by the Government of Catalonia (Comissionat per a Universitats i Recerca, Departament d’Innovació,
Acknowledgments:
This work was partially funded by Indra, the University of Lleida, the RecerCaixa 2013
Universitats i Empresa) and the European Social Fund (ECO/1794/2015).
grant, and by the Government of Catalonia (Comissionat per a Universitats i Recerca, Departament d’Innovació,
Universitats
i Empresa) and
the Moreno,
European
Social Clotet,
Fund (ECO/1794/2015).
Author Contributions:
Javier
Eduard
Ruben Lupiañez, Marcel Tresanchez and Dani Martínez
designed
the
holonomic
motion
system
and
also
designed
performed
theTresanchez
experiments;
Tomàs
Author Contributions: Javier Moreno, Eduard Clotet,
Ruben and
Lupiañez,
Marcel
and
Dani Pallejà,
Martínez
Jordi
Casanovas
and
Jordi
Palacín
supervised
the
experiments;
Eduard
Clotet
and
Jordi
Palacín
analyzed
the
designed the holonomic motion system and also designed and performed the experiments; Tomàs Pallejà,
data;
Javier Moreno
andPalacín
Jordi Palacín
wrote the
the experiments;
paper.
Jordi
Casanovas
and Jordi
supervised
Eduard Clotet and Jordi Palacín analyzed the data;
Javier Moreno and Jordi Palacín wrote the paper.
Conflicts of Interest: The authors declare no conflict of interest.
Conflicts of Interest: The authors declare no conflict of interest.
Sensors 2016, 16, 1658
20 of 21
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