Inter
national
J
our
nal
of
Robotics
and
A
utomation
(IJRA)
V
ol.
9,
No.
3,
September
2020,
pp.
153
159
ISSN:
2089-4856,
DOI:
10.11591/ijra.v9i3.pp153-159
r
153
F
orward
kinematic
analysis
of
Dobot
using
closed-loop
method
J
a
vier
Sanjuan,
Mohammad
Rahman,
Iv
an
Rulik
Mechanical
Engineering
Department,
Uni
v
ersity
of
W
isconsin-Mil
w
auk
ee,
United
States
Article
Inf
o
Article
history:
Recei
v
ed
Oct
23,
2019
Re
vised
Jan
12,
2020
Accepted
Mar
4,
2020
K
eyw
ords:
Dobot
Dynamical
analysis
F
orw
ard
kinematics
P
arallel
robots
Serial
robots
ABSTRA
CT
Dobot
is
a
h
ybrid
robot
that
combines
features
from
parallel
and
serial
robots.
Because
of
this
characteristic,
the
robot
e
xcels
for
is
reliability
,
allo
wing
its
implementation
in
di
v
erse
applicati
ons.
Therefore,
researchers
ha
v
e
studied
its
kinematics
to
impro
v
e
its
capabilities.
Ho
we
v
er
,
to
the
e
xtent
of
our
kno
wledge,
no
analysis
has
been
reported
taking
into
consideration
the
closed-loop
configuration
of
Dobot.
Thus,
this
article
presents
the
complete
analytical
solution
for
the
forw
ard
kinematics
of
Dobot,
considering
each
link.
The
results
are
e
xpected
to
be
utili
zed
in
the
de
v
elopment
of
a
dynamical
model
that
contemplates
the
dynamics
of
each
element
of
the
robot.
This
is
an
open
access
article
under
the
CC
BY
-SA
license
.
Corresponding
A
uthor:
Ja
vier
Sanjuan,
Biorobotics
Lab,
USR
281,
Uni
v
esirty
Services
and
Research
Building,
UW
-Mil
w
auk
ee,
115
East
Reindl
W
ay
,
Glendale,
WI,
53212,
United
States.
Email:
jsanjuan@uwm.edu
1.
INTR
ODUCTION
Dobot
is
a
3
De
gree
of
Freedom
(DOF)
type
of
commercial
serial
manipulator
that
posses
numerous
attracti
v
e
features
bec
ause
of
its
h
ybrid
serial
and
parallel
configuration.
One
of
its
principal
characteristics
is
the
preci
sion
due
to
Dobot
uses
a
four
-bar
linkage
to
actuate
the
rotation
of
each
link.
Additionally
,
Dobot
utilizes
a
parallelogram
mechanism
to
ensure
that
the
orientation
of
the
end-ef
fector
remains
constant.
Therefore,
because
of
these
features
and
its
lo
w
cost,
v
arious
researches
ha
v
e
implemented
Dobot
in
di
v
erse
applications
such:
3D
printing
[1],
electrochemical
writing
[2,
3],
Pick
and
place
[4–8],
education
[9],
sur
gery
[10],
and
e
v
en
electric
v
ehicle
char
ging
[11].
Because
of
the
di
v
erse
applications
of
Dobot,
researches
ha
v
e
conducted
dif
ferent
types
of
analys
is
on
this
robot.
T
.
Cheng
et
al.
analyzed
the
forw
ard
kinematics,
in
v
erse
kinematics,
and
jacobian
matrix
of
Dobot
[11].
Moreo
v
er
,
O.
Hock,
et
al.
included
an
analysis
of
the
pseudoin
v
erse
method
to
compute
the
in
v
erse
kinematics
of
Dobot
numerically
[12].
G.
Y
u
et
al.
modified
the
end-ef
fector
of
Dobot
to
included
a
gripper
and
a
camera,
to
implement
hand-e
ye
calibration
[13].
Ho
we
v
er
,
to
the
e
xtent
of
our
kno
wledge,
no
research
has
conducted
the
forw
ard
kinematics
analysis
of
the
linkage
that
composes
the
robot.
The
forw
ard
kinematics
analysis
permits
to
obtain
the
kinematic
of
each
link,
which
is
a
requirem
ent
to
de
v
elop
a
detail
dynami
cal
model
of
Dobot.
This
type
of
model
allo
ws
the
implementation
of
rob
ust
control
strate
gies
such
as
in
v
erse
dynamics
control
[14,
15],
permitting
Dobot
to
be
used
in
the
application
of
high
v
elocity
while
maintaining
its
precision
[16].
V
arious
researcher
has
w
ork
ed
in
the
computation
of
the
direct
kinematics
for
dif
ferent
parallel
robots.
Y
ujiong
L.,
et
al.
presented
the
computation
of
the
forw
ard
kinematics
of
an
H4
parallel
robot
using
a
geometric
approach
[17].
Olaru
D.
obtained
the
forw
ard
and
in
v
erse
J
ournal
homepage:
http://ijr
a.iaescor
e
.com
Evaluation Warning : The document was created with Spire.PDF for Python.
154
r
ISSN:
2089-4856
kinematics
of
a
5DOF
robot,
between
the
rele
v
ant
results
from
this
research
is
the
increasing
of
the
robot
precision
due
to
the
implementation
of
the
computation
of
the
forw
ard
kinematics
in
its
control
[18].
Jin
S.
K.,
et
al.
implemented
the
kinematic
analysis
of
a
4DOF
parallel
robots
for
MRI-Guided
percutaneous
interv
en-
tions
[19].
T
ang
et
al.
study
the
kinematics
of
a
no
v
el
2R1T
parallel
mechanism
[20].
Therefore,
because
of
the
adv
antages
of
a
complete
kinematical
model,
this
research
focuses
on
the
computation
of
the
forw
ard
kine-
matics
of
Dobot,
considering
the
closed-loop
configuration
of
each
actuated
joint.
The
results
of
the
algorithm
were
v
alidated
using
a
CAD
model,
v
erifying
the
accurac
y
of
the
results.
The
or
g
anization
of
the
article
is
the
follo
wing:
Section
2
describes
the
principal
joints
of
Dobot.
Section
3
presents
the
ki
nematical
diagram
of
Dobot,
and
e
xplain
the
relation
between
links.
Section
4
presents
the
computation
of
the
forw
ard
kinematic
of
Dobot.
Section
5
compares
the
results
of
the
obtained
equations
with
a
CAD
model
of
Dobot,
to
v
erify
the
accurac
y
of
the
results.
Figure
1.
3D
model
representation
of
DOBO
T
2.
DOBO
T
DESCRIPTION
The
cad
model
of
Dobot
is
presented
in
Figure
1.
As
can
be
seen,
joints
1
and
2
(
q
1
&
q
2
)
actuates
in
the
same
direction,
therefore,
the
actuation
of
those
joints,
and
their
ef
fect
on
the
kinematics
of
the
robot
can
be
analyzed
with
a
plane
model.
On
the
other
hand,
joint
3
rotates
the
robot
in
the
y
direction,
which
changes
the
orientation
Dobot.
Since
this
rotation
is
pe
rpendicular
to
the
other
actuated
joints,
this
actuation
is
not
considered
in
the
analysis.
Also,
T
.
Cheng
et
al.
presented
a
model
to
study
the
influence
of
this
joint
in
the
orientation
of
Dobot
[11].
In
Figure
2
the
sectional
vie
w
sho
ws
the
mechanism
inside
DOBO
T’
s
case,
where
the
v
ariables
e
xplained
in
Section
2
can
be
related
using
Figures
2
and
3.
Figure
2.
Sectional
vie
w
of
DOBO
T
3.
KINEMA
TICS
MODEL
T
o
f
acilitate
the
analysis
of
Dobot
the
kinematic
diagram
that
presents
all
the
links
is
obtained,
as
sho
wn
in
Figure
3.
Note
that
the
actuate
d
angles
are
represented
by
q
1
and
q
2
.
Additionally
,
since
dobot
posses
a
parallel
configuration,
the
actuation
of
the
second
joint
is
obtained
by
a
four
bar
mechanism
composed
for
links:
L
q
2
;
L
7
;
and
L
6
,
that
transmit
the
mo
v
ement
from
the
base
to
this
joint.
The
end
ef
fector
is
represent
by
Int
J
Rob
&
Autom,
V
ol.
9,
No.
3,
September
2020
:
153
–
159
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Rob
&
Autom
ISSN:
2089-4856
r
155
point
P
,
the
orientation
of
this
link
is
not
directly
actuated,
and
is
dictated
by
a
W
att’
s
mechanism
[21]
that
is
composed
for
links:
L
2
;
L
3
;
L
4
and
L
5
.
Then,
to
v
erify
that
the
kinematic
representation
of
Dobot
is
correct,
the
Gruebler
equation
[22]
is
used
to
compute
the
DOF
of
the
system,
as
follo
ws:
Figure
3.
Kinematics
model
N
=
3
L
2
J
(1)
Where
N
is
the
number
of
DOF;
L
is
the
number
of
links,
and
J
is
the
number
of
joints.
Therefore,
the
number
of
DOF
is
2,
which
is
the
e
xpected
result.
4.
FOR
W
ARD
KINEMA
TICS
T
o
obtain
the
forw
ard
kinematics
it
is
used
the
closed-loop
method
[23].
This
method
requires
the
analysis
of
closed-loop
v
ector
to
obtain
the
position
equati
ons.
The
number
of
closed-equations
is
determined
by
the
use
of
the
relation
for
Li
et
al.
in
[17],
as
follo
ws:
d
=
J
L
=
3
(2)
Therefore,
it
is
required
to
use
three
closed-loop
equations
to
obtain
the
position
of
Dobot.
The
closed-loop
used
are
presented
in
Figure
4.
Figure
4.
Closed-loops
diagrams
F
or
the
first
loop,
the
obtained
equation
is
the
follo
wing:
~
L
q
2
+
~
L
7
+
~
L
63
=
~
L
0
+
~
L
q
1
(3)
Equation
(3)
is
re
grouped
in
terms
of
L
6
and
e
xpressed
in
v
ector
form,
as
follo
ws:
L
63
cos
6
+
3
sin
6
+
3
=
L
0
cos
0
sin
0
+
L
q
1
cos
q
1
sin
q
1
L
q
2
cos
q
2
sin
q
2
L
7
cos
7
sin
7
(4)
F
orwar
d
kinematic
analysis
of
Dobot
using
closed-loop
method
(J
avier
Sanjuan)
Evaluation Warning : The document was created with Spire.PDF for Python.
156
r
ISSN:
2089-4856
Then,
adding
the
square
of
the
x
side
and
the
square
of
the
y
side,
the
term
corresponding
to
6
is
simplified.
Thus,
obtaining
an
e
xpression
that
depends
on
7
;
q
1
;
q
2
;
and
0
.
Ho
we
v
er
,
since
q
1
and
q
2
are
entries
of
the
system,
and
0
is
a
fix
angle,
the
only
v
ariable
of
interest
is
7
.
Therefore,
the
resultant
e
xpression
is
e
xpressed
in
term
of
7
as
follo
ws:
A
1
cos
7
+
A
2
sin
7
+
A
3
=
0
(5)
Where
the
terms
A
i
are
parameters
that
depends
on
the
dimensions
of
Dobot,
and
kno
wn
angles.
Then,
using
the
half
tangent
substitution
tan
7
2
=
c
1
,
with
cos
7
=
1
c
2
1
1+
c
2
1
and
sin
7
=
2
c
1
1+
c
2
1
,
(5)
is
reduced
to
an
algebraic
e
xpression,
as
follo
ws:
k
1
C
2
2
+
k
2
C
2
+
k
3
=
0
(6)
Then,
using
the
quadractic
equation
in
(6),
the
follo
wing
equation
is
obtained:
C
1
=
k
2
p
k
2
2
4
k
1
k
3
2
k
1
(7)
Replacing
c
1
for
the
tan
7
2
,
the
e
xpression
for
7
is
obtained:
7
=
atan
0
@
p
2
q
L
q
1
2
L
q
2
2
(cos
(2
q
1
2
q
2
)
1)+2
L
q
1
2
sin
(
q
1
)
2
L
q
1
L
q
2
sin
(
q
2
)
2
L
q
1
(L
q
1
+
L
q
1
cos
(
q
1)
L
q
2
cos
(
q
2
)
L
q
2
cos
(
q
1
q
2
))
1
A
(8)
No
w
that
the
e
xpression
for
7
is
obtained,
6
is
computed
from
applying
the
dot
product
of
4
with
unitary
v
ector
^
i
=
[1
;
0]
T
,
and
solving
for
6
,
generating:
6
=
acos
L
q
2
cos
(
q
2
)
L
q
1
cos
(
q
1
)
+
L
q
1
cos
(
7
)
L
q
2
(9)
Analogously
,
the
second
loop
equation
is
presented
in
the
follo
wing:
~
L
1
+
~
L
2
+
~
L
31
=
~
L
q
2
+
~
L
7
+
~
L
61
(10)
The
tw
o
v
ariables
to
solv
e
from
(10)
are
2
and
3
,
whose
solutions
are
obtained
applying
the
same
methods
for
7
and
6
,
respecti
v
ely
.
Therefore,
the
solution
for
3
is
presented
belo
w:
3
=
2atan
k
12
p
k
2
12
4
k
11
k
13
2
k
11
!
(11)
The
v
alues
of
k
11
;
k
12
;
and
k
13
are
presented
in
the
Appendix
section.
The
solution
for
2
is
presented
ne
xt.
2
=
acos
L
q
2
cos
(
1
+
6
)
L
q
2
cos
(
1
+
3
)+L
q
2
cos
(
q
2
)
L
q
2
cos
(
1
)
+
L
q
1
cos
(
7
)
L
q
1
(12)
Then,
using
the
same
procedure
as
before,
the
third
loop
equation
is
the
follo
wing:
~
L
3
+
~
L
4
=
~
L
62
+
~
L
51
(13)
From
solving
(13),
the
solution
for
5
is
the
follo
wing:
5
=
2
atan
0
@
p
2
q
L
4
2
L
q
2
2
(cos
(2
3
2
6
)
1)
2
L
q
2
2
sin
(
3
)
+
2
L
4
L
q
2
sin
(
6
)
2
L
q
2
(L
q
2
L
4
cos
(
6
)
+L
q
2
cos
(
3
)
L
4
cos
(
3
6
))
1
A
(14)
And
for
4
:
4
=
acos
L
q
2
cos
(
1
+
5
)
+
L
4
cos
(
6
)
L
q
2
cos
(
3
)
L
4
(15)
Int
J
Rob
&
Autom,
V
ol.
9,
No.
3,
September
2020
:
153
–
159
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Rob
&
Autom
ISSN:
2089-4856
r
157
Lastly
,
using
the
fourth
loop
as
presented
in
Figure
5,
the
v
alues
of
x
and
y
are
obtained
using
the
follo
wing
e
xpression:
L
q
1
cos
(
q
1
)
+
L
4
cos
(
6
)
+
L
5
cos
(
5
)
=
x
L
q
1
sin
(
q
1
)
+
L
4
sin
(
6
)
+
L
5
sin
(
5
)
=
y
(16)
Note
that
(8),
(9),
(11),
(12),
(14),
(15),
and
(16)
were
simplified
considering
the
dimensional
parameters
presented
in
T
able
1.
Figure
5.
F
ourth
loop
T
able
1.
Dimensional
parameters
on
DOBO
T
P
arameter
V
alues
L
63
,
L
q
2
,
L
1
,
L
31
,
L
61
,
L
3
,
L
51
43
[mm]
L
7
,
L
q
1
,
L
2
135
[mm]
L
4
,
L
62
147
[mm]
L
0
0
[mm]
0
,
1
,
2
,
3
0
[rad]
1
2
:
66
[rad]
1
5
:
1
[rad]
1
0
:
7
[rad]
5.
RESUL
TS
T
o
v
alidate
the
kinematics
model
and
e
v
aluate
its
accurac
y
,
a
comparison
w
as
done
using
the
CAD
obtained
from
D
O
B
O
T
©
.
Both,
the
results
from
the
CAD
model
and
the
Kinematic
model
are
presented
in
T
able
.
When
comparing
the
results
it
is
clear
that
the
computed
v
alues
and
the
CAD
model
present
the
same
results.
Therefore,
the
forw
ard
kinematic
analysis
is
accurate.
T
able
2.
T
est
of
the
Kinematic
Model
V
ariables
CAD
measur
e
[
o
]
Kinematic
Model
[
o
]
q
1
44
44
q
2
150
150
2
44
44
3
40.28
40.2823
4
30
30
5
0.28
0.2823
6
30
30
7
44
44
6.
CONCLUSIONS
This
paper
presents
the
forw
ard
kinematic
analys
is
of
Dobot,
a
3DOF
h
ybrid
robot
that
is
posses
both,
a
serial
and
parallel
configuration.
The
equations
were
e
xpressed
using
the
closed-loop
method
and
were
solv
ed
analytically
by
the
application
of
the
tangent
half-angle
substitution.
This
result
is
e
xpected
to
be
utilized
in
the
computation
of
the
dynamics
of
Dobot,
to
implement
elaborated
control
strate
gies
such
as
in
v
erse
dynamic
control.
The
accurac
y
of
the
equations
w
as
also
v
alidated
using
a
CAD
model,
obtaining
an
easy
to
utilize
a
set
of
equations
to
obtain
the
kinematic
of
Dobot
as
a
function
of
the
joint
angles.
F
orwar
d
kinematic
analysis
of
Dobot
using
closed-loop
method
(J
avier
Sanjuan)
Evaluation Warning : The document was created with Spire.PDF for Python.
158
r
ISSN:
2089-4856
APPENDIX
The
follo
wing
equations
present
the
v
alues
of
k
11
,
k
12
,
and
k
13
.
k
11
=2L
q
2
2
cos
(
1
q
2
)
2L
q
2
2
cos
(
1
1
)
+
2L
q
2
2
cos
(
1
6
)
2L
q
2
2
cos
(
q
2
1
)
+
2L
q
2
2
cos
(
q
2
6
)
2L
q
2
2
cos
(
1
6
)
+
4L
q
2
2
2L
q
1
L
q
2
cos
(
1
7
)
+
2L
q
1
L
q
2
cos
(
6
7
)
+
2L
q
1
L
q
2
cos
(
1
7
)
+
2L
q
1
L
q
2
cos
(
q
2
7
)
(17)
k
12
=4
L
q
2
2
sin
(
1
q
2
)
4
L
q
2
2
sin
(
1
1
)
+
4
L
q
2
2
sin
(
1
6
)
+
4
L
q
1
L
q
2
sin
(
1
7
)
(18)
k
13
=2L
q
2
2
cos
(
1
1
)
2L
q
2
2
cos
(
1
q
2
)
2Lq2
2
cos
(
1
6
)
2L
q
2
2
cos
(
q
2
t
1
)
+
2L
q
2
2
cos
(
q
2
t
6
)
2L
q
2
2
cos
(
t
1
t
6
)
+
4L
q
2
2
2L
q
1
L
q
2
cos
(
1
7
)
+
2L
q
1L
q
2
cos
(
6
7
)
2L
q
1
L
q
2
cos
(
1
7
)
+
2L
q
1
L
q
2
cos
(
q
2
7
)
(19)
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BIOGRAPHIES
OF
A
UTHORS
J
a
vier
Sanjuan
is
a
Ph.D.
student
at
the
BioRobotics
Lab,
Uni
v
ersity
of
W
isconsin-Mil
w
auk
ee.
He
obtained
his
masters
and
bachelors
de
gree
s
in
Mechanical
Engineering
at
Uni
v
ersidad
del
Norte,
Cololmbia
(2012-2016).
His
research
interest
includes
dynamics,
control,
and
parallel
robots.
In
Mechanisms
and
Machine
Theory
Journal,
IEEE
Latin
America,
and
other
scientific
publications,
he
has
serv
ed
as
in
vited
re
vie
wer
.
Besides,
he
is
one
of
the
founders
of
the
V
ertical
Inte
grated
Projects
(VIP)
student
association
in
Uni
v
ersidad
del
Norte,
Colombia.
Further
info
on
his
homepage:
https://www
.researchg
ate.net/profile/Ja
vier
S
anj
uan
5
Mohammad
Rahman
is
with
the
Mechanical
and
Biomedical
Engineering
Department,
Uni
v
ersity
of
W
isconsin-Mil
w
auk
ee,
WI,
USA.
As
Director
of
the
BioRobotics
Lab
at
the
Uni
v
ersity
of
W
isconsin-Mil
w
auk
ee,
he
brings
the
resources
and
e
xpertise
of
an
interdisciplinary
R&D
team.
F
or
more
than
15
years
he
has
been
researching
bio-mechatronics/bio-robotics
with
emphasis
on
the
design,
de
v
elopment
and
control
of
wearable
robots
to
rehabilitate
and
assis
t
elderly
and
ph
ysically
disabled
indi
viduals
who
ha
v
e
lost
their
upper
-limb
function
or
motion
due
to
strok
e,
cardio
v
ascular
disease,
trauma,
sports
injuries,
occupational
injuries,
and
spinal
cord
injuries.
He
recei
v
ed
a
BSc
Engineering
(mechanical)
de
gree
from
Khulna
Uni
v
ersity
of
Engineering
&
T
echnology
,
Bangladesh
in
2001,
a
Master
of
Engineering
(bio-robotics)
de
gree
from
Sag
a
Uni
v
ersity
,
Japan
in
2005
and
a
PhD
in
Engineering
(bio-robotics)
from
´
Ecole
de
technologi
e
sup
´
erieure
(ETS),
Uni
v
ersit
´
e
du
Qu
´
ebec,
Canada
in
2012.
He
w
ork
ed
as
a
postdoctoral
resea
rch
fello
w
in
the
School
of
Ph
ysical
&
Occupational
Therap
y
,
McGill
Uni
v
ersity
(2012-2014).
His
research
interests
are
in
bio-robotics,
e
xosk
eleton
robot,
intelligent
system
and
control,
mobile
robotics,
nonlinear
control,
control
using
biological
signal
such
as
electromyogram
signals.
Iv
an
Rulik
is
a
senior
Electrical
and
Biomedical
Engineering
international
student
from
Uni
v
ersidad
de
Colombia,
and
will
get
his
de
gree
in
2020.
He
is
currently
a
Research
A
ssistant
in
the
Mechanical
Engineering
Department
at
Uni
v
ersi
ty
of
W
isconsin-Mil
w
auk
ee,
w
orking
in
the
BioRobotics
lab
.
He
brings
e
xpertise
in
programming
using
Matlab
and
designing
in
CAD
with
In
v
entor
,
AutoCAD
and
Solidw
orks.
His
research
interests
include
robotics,
microrobotics,
medicine,
rehabilitation
de
vices
and
electromagnetic
manipulation
at
small
scales
He
is
af
filiated
with
IEEE
as
student
member
.
Besides,
he
is
also
in
v
olv
ed
in
the
IEEE
Industry
Applications
Society
Student
Branch
from
Uni
v
ersidad
de
los
Andes,
Colombia.
F
orwar
d
kinematic
analysis
of
Dobot
using
closed-loop
method
(J
avier
Sanjuan)
Evaluation Warning : The document was created with Spire.PDF for Python.