Indonesian
J
our
nal
of
Electrical
Engineering
and
Computer
Science
V
ol.
25,
No.
3,
March
2022,
pp.
1297
∼
1307
ISSN:
2502-4752,
DOI:
10.11591/ijeecs.v25.i3.pp1297-1307
❒
1297
The
measur
ement
set
r
epr
esentation
of
the
body
postur
e
based
on
gr
oup
theory
Changjian
Deng,
Xiaona
Xie
School
of
Control
Engineering,
Chengdu
Uni
v
ersity
of
Information
T
echnology
,
Chengdu,
China
Article
Inf
o
Article
history:
Recei
v
ed
Jun
22,
2021
Re
vised
Dec
20,
2021
Accepted
Jan
11,
2022
K
eyw
ords:
Attitude
sensor
Body
posture
Group
theory
Lie
group
Representation
ABSTRA
CT
The
foundation
of
measurement
is
the
representation
of
measurement
set.
The
paper
proposes
a
body
posture
measurement
set
representation
method
based
on
concepts
of
group,
elds,
and
ring.
It
attempts
to
e
xplore
the
intrinsic
relationship
among
numer
-
ous
dif
ferent
measurement
set.
The
attitude
sensor
is
used
to
measure
the
attitude,
and
the
measurement
set
representation
and
processing
are
analyzed
based
on
the
lie
group
theory
in
the
paper
.
In
the
paper
,
the
paper
maps
body
posture
data
into
image,
and
it
is
easier
to
identify
the
error
of
posture
measurement
data.
Meanwhile,
the
simulation
and
test
results
sho
w
that
the
method
can
represent
body
posture
data
and
detect
it’
s
defects
easily
.
This
is
an
open
access
article
under
the
CC
BY
-SA
license
.
Corresponding
A
uthor:
Changjian
Deng
School
of
Control
Engineering,
Chengdu
Uni
v
ersity
of
Information
T
echnology
No.
24,
Block
1,
Xuefu
Road,
Chengdu
City
,
China
Email:
chenglidcj@cuit.edu.cn
1.
INTR
ODUCTION
Normally
,
measurement
refers
to
a
group
of
operations
to
determine
the
“quantity
v
alue”.
The
pur
-
pose
of
measurement
is
to
mak
e
the
quantity
information
contained
in
the
object
ob
vious
and
comparable.
In
comple
x
and
specic
application,
the
measured
quantity
may
no
longer
be
a
scattered
string
of
irrele
v
ant
data.
These
data
and
their
changes
may
be
manifolds
or
groups.
Exploring
these
rela
tionships
is
an
important
method
to
ensure
the
accurac
y
of
measurement
and
control.
This
paper
mainly
studies
this
problem.
The
de-
v
elopment
of
group
theory
pro
vides
a
basis
for
these
studies.
No
w
adays,
group
theory
has
man
y
applications
[1].
F
or
e
xample,
it
is
also
a
po
werful
tool
in
signal
processing
[2],
[3],
signal
representation
[4],
and
me-
chanics
[5].
This
paper
is
mainly
concerned
with
the
representation
of
group
theory
.
The
related
research
of
representation
includes
tw
o
catalogues,
one
is
signal
representation
and
detection
[6]-[9],
and
the
other
is
deep
learning
and
object
tracking
[10]-[12].
The
paper
focuses
on
measurement
set
representation
based
on
group
theory
.
The
contrib
utions
of
paper
include:
1)
it
proposes
image
representation
method;
2)
it
presents
body
posture
representation
and
analysis
method
based
on
lie
group.
The
paper
is
or
g
anized
as
follo
ws:
in
section
2,
we
present
the
related
w
ork.
In
section
3,
we
present
the
measurement
sets
and
group
representation.
Section
4
focusi
ng
on
measurement
set
map.
The
e
xperiments
and
simulation
are
presented
in
section
5.
The
section
6
is
conclusion.
J
ournal
homepage:
http://ijeecs.iaescor
e
.com
Evaluation Warning : The document was created with Spire.PDF for Python.
1298
❒
ISSN:
2502-4752
2.
MEASUREMENT
SETS
AND
GR
OUP
REPRESENT
A
TION
2.1.
Fields
of
measur
ement
The
measurement
processing
often
is
undertak
en
in
dif
ferent
domains,
for
e
xample:
the
time
-frequenc
y
electronic
magnetic
domain;
the
tensor
mechanical
domain;
the
thermotical
domain;
ener
gy
and
mass
domain;
time
and
space
domain;
and
so
on.
In
nature,
most
quantity
of
entity
to
be
measured
is
represented
by
signal
as
signal
processing
is
algebraic
in
nature.
The
measurement
processing
is
algebraic
in
nature:
−
The
real
v
oltage
signal
set
V
in
time
domain
is
a
group.
−
The
real
v
oltage
signal
set
V
in
time
domain
is
a
ring.
−
The
real
v
oltage
signal
set
V
in
time
domain
is
a
eld.
Generally
,
measurable
sets
normally
itself
and
its
map
are
elds.
Domain
is
structured
sets.
In
group
analysis,
lagrange
theorem
and
homomorphism
principle
in
group
theory
are
the
basis
of
the
analysis
of
measurement
set.
2.2.
Continue
medium
measur
ement
sets
The
measurement
set
may
come
from
dif
ferent
time
test,
dif
ferent
space
test,
or
dif
ferent
circ
u
m
stance
test.
In
a
ri
gid
body
,
architecture,
material,
quantity
form
their
dif
ferent
part
and
their
moti
v
ation
form
a
continue
measurement
sets.
Here
we
discuss
tw
o
kinds
of
problem.
The
y
may
be
useful
in
sensor
,
moti
v
ation
analysis,
and
so
on.
2.2.1.
The
rst
kind
of
pr
oblem
is
perf
ormance
analysis
and
defect
detection
of
material
in
sensor
tech-
nology
F
or
e
xample,
the
analysis
the
dielectric
coef
cient,
piezoelectric
coef
cient,
piezo-resistance
coef
-
cient,
and
so
on.
There
are
some
related
softw
are
e
xamples
of
group
in
engineering
applicati
on.
Puschel
and
Moura
[13]
proposed
the
ISO
TR
OPY
softw
are
tools,
it
collections
softw
are
which
applies
group
theoretical
methods
to
the
analysis
of
phase
transitions
in
crystalline
solids.
Method
of
analysis
the
measurement
related
coef
cient:
as
an
introduction,
here
gi
v
e
an
e
xample
of
dielectric
coef
cient
of
cubic
dielectric
capacitance
sensor
.
And
it
can
be
pro
v
ed
that
this
coef
cient
is
scalar
quantity
.
Pro
v
e
[14]:
assume
the
tensor
of
dielectric
coef
cient
is
ϵ
αβ
(
α
,
β
=
1
,
2
,
3
),
then:
D
α
=
P
3
β
=1
ϵ
αβ
E
β
select
axis,
let
E
=
E
j
D
α
=
P
α,β
ϵ
αβ
E
δ
β
2
=
ϵ
β
2
E
=
E
αy
E
(
α
=
x,
y
,
z
)
let
axis
‘y’
as
basis
axis
of
rotation,
do
C
1
4
operation:
z
→
z
′
=
x,
x
→
x
′
=
−
z
,
then,
D
′
x
=
D
z
,
D
′
z
=
−
D
x
.
,
with
the
in
v
arity
property
of
C
4
,
so,
D
′
x
=
D
x
,
D
′
z
=
D
z
.
,
then
we
get:
ϵ
xy
=
ϵ
z
y
ϵ
xy
=
−
ϵ
z
y
so,
ϵ
xy
=
ϵ
z
y
=
0
,
and
it
is
same
as:
ϵ
xz
=
ϵ
y
z
=
ϵ
y
x
=
ϵ
z
x
=
0
then
let
E
along
the
diagonal
of
a
cube
(
C
3
axis),
then:
E
x
=
E
y
=
E
z
=
1
√
3
E
,
then:
Indonesian
J
Elec
Eng
&
Comp
Sci,
V
ol.
25,
No.
3,
March
2022:
1297–1307
Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian
J
Elec
Eng
&
Comp
Sci
ISSN:
2502-4752
❒
1299
D
x
=
1
√
3
ϵ
xx
E
D
y
=
1
√
3
ϵ
y
y
E
D
z
=
1
√
3
ϵ
z
z
E
do
C
1
3
and
C
2
3
operation,
and
consider
the
in
v
arity
of
C
3
.
Then
D
x
=
D
y
=
D
z
and
ϵ
xx
=
ϵ
y
y
=
ϵ
z
z
=
ϵ
,
and
get
the
solution.
2.2.2.
The
another
kind
of
pr
oblem
is
faults
or
distortion
caused
by
material
defects,
the
important
tool
is
homotr
opy
gr
oup
No
w
adays,
point
defect
research
in
semic
on
duct
ors
has
g
ained
remarkable
ne
w
momentum
due
to
the
identication
of
special
point
defects
that
can
implement
qubits
and
single
photon
emitters
with
unique
characteristics.
The
defect
of
continuum
can
be
represented
using
homotrop
y
group.
Homotrop
y
groups
in
order
parameter
space
T
=
G/H
are
used
to
represent
and
classify
defects.
The
union
of
defects
is
obtained
by
multiplication
of
homotrop
y
groups.
In
the
plane
ordered
space
eld
T
,
the
winding
number
n
is
an
importa
nt
parameter
.
F
or
the
images
with
the
same
winding
number
n,
the
y
can
be
changed
into
the
same
features
by
local
adjustment.
And
the
kind
of
defects
wi
th
same
n
can
be
transformed
into
each
other
,
which
is
barrier
free
in
topology
.
The
y
are
same
homotrop
y
group.
Method
of
analysis
the
defect
of
planar
spin
system,
planar
spin
system
[14]:
Its
Order
parameter:
S
=
i
cosθ
+
j
sinθ
T
ransformation
group:
G=T(1);
T
ϕ
(
θ
)
→
θ
−
ϕ
Isotropic
group:
H=
T
aπ
(
n
)
,
n
=
0
,
±
1
,
±
2
,
...,
Discrete
group:
H
0
=
e
.
So,
Q
1
(
T
)
=
H
=
Z
,
this
is
additi
v
e
group
of
inte
gers.
2.3.
Measur
ement
sets
r
epr
esentation
method
As
sho
wn
in
T
able
1,
dif
ferent
measurement
processing
has
dif
ferent
method
to
represent
their
p
a
ram-
eters.
Some
rules
are
:
a.)
P
arameters
(measurement
sets)
can
represent
the
main
property
of
object
visually
,
mapping
them
into
graph-
based
data
though
positi
v
e
and
ne
g
ati
v
e
direction
calculation
(mapping).
T
able
1
gi
v
e
some
e
xist
method
and
its
e
xtending.
An
e
xample
is
sho
wn
in
Figure
1.
b
.)
This
graph-based
measurement
set
should
be
measurable
and
has
some
dened
mathematical
fe
ature
(or
Algebraic).
T
able
2
is
an
e
xample
of
body
posture
measurement
set
and
its
group
representation.
T
able
1.
P
arameters
representation
Properties
Graph
method
Infra
thermo-image
Pix
el
temperature
map
to
image
Nature
sound
T
ime-frequenc
y
mapping
Automoti
v
e
radar
Geometry
parameters
mapping
Soil
temperature
of
dif
ferent
place
T
emperature
map
to
image
Figure
1.
The
diagram
of
measurement
set
map
The
measur
ement
set
r
epr
esentation
of
the
body
postur
e
based
on
gr
oup
theory
...
(Changjian
Deng)
Evaluation Warning : The document was created with Spire.PDF for Python.
1300
❒
ISSN:
2502-4752
T
able
2.
P
arameters
measurable
representation
P
arameters
Algebraic
method
Acceleration
and
Gyroscope
signal
data
set
Perfect
linear
matrix
group
Eulerian
angle
measurement
data
Perfect
linear
matrix
group
Defect
data
Homotrop
y
group
When
we
map
these
data
into
image,
it
is
easy
to
see
some
defect
of
sensor
or
measurement
error
.
The
Figure
2
is
acc
data
of
three
dif
ferent
sensor
,
the
Figure
3
is
their
gyro
data.
The
measurement
en
vironment
is
static.
So,
it
is
v
ery
clear
that
there
is
something
wrong
in
sensor
2.
And
this
is
true
case.
W
e
use
Kalman
lter
and
other
methods
to
deal
with
the
acc
data
and
gyro
data,
then
we
get
the
Eulerian
angle
data.
It
is
sho
wn
in
Figure
4.
Figure
2.
The
acc
measurement
set
mapping
Figure
3.
The
h
yro
measurement
set
mapping
Indonesian
J
Elec
Eng
&
Comp
Sci,
V
ol.
25,
No.
3,
March
2022:
1297–1307
Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian
J
Elec
Eng
&
Comp
Sci
ISSN:
2502-4752
❒
1301
Figure
4.
The
Eulerian
angle
data
set
mapping
In
planar
operat
ion,
the
rigid
status
is
dened
by
the
angle
of
the
joint.
In
n-dimensional
real
space,
i
t
is
S
O
(2)
×
S
O
(2)
...
×
S
O
(2)
.
In
this
Algebraic
structure,
the
group
operators
are
dene
d
based
on
addition
operation.
Group
addition
operation:
q
1
,
q
2
∈
ℜ
n
,
q
1
⊕
q
2
=
q
1
+
q
2
zero
element:
E
=
0
=
[0
,
0
,
...,
0]
T
in
v
erse
operation:
q
∈
ℜ
n
,
⊖
q
=
−
q
scalar
product:
α
∈
ℜ
,
q
∈
ℜ
n
,
α
⊙
q
=
q
⊕
q
⊕
...
⊕
q
=
α
·
q
space
rigid
body
posture
often
is
dened
by
3
×
3
matrix,
the
group
operator
is
dened
by
the
matrix
product,
that
is
SO(3).
Group
addition
operation:
R
1
,
R
2
∈
S
O
(3)
,
R
1
⊕
R
2
=
R
1
R
2
zero
element:
E
=
I
3
=
diag
([1
,
1
,
1])
in
v
erse
operation:
R
∈
S
O
(3)
,
⊖
R
=
R
−
1
scalar
product:
α
∈
ℜ
,
R
∈
S
O
(3)
,
α
⊙
q
=
R
⊕
R
⊕
...
⊕
R
=
R
·
R
...
·
R
.
The
measur
ement
set
r
epr
esentation
of
the
body
postur
e
based
on
gr
oup
theory
...
(Changjian
Deng)
Evaluation Warning : The document was created with Spire.PDF for Python.
1302
❒
ISSN:
2502-4752
3.
MEASUREMENT
SET
MAP
3.1.
Fr
om
ODE
(ordinary
differ
ential
equation)
to
gr
oup
In
sensor
technology
,
the
polynomials
is
used
to
stand
for
the
standalone
static
characteristics
of
the
sensor
system
and
ordi
nary
dif
ferential
equation
is
used
to
stand
for
the
dynamic
feature
of
t
he
linear
time
in-
v
ariant
sensor
system.
More
generally
,
the
state
space
equation
is
used
to
stand
for
the
system
motion
equation.
Polynomials
are
created
from
names,
inte
gers,
and
other
v
alues
using
the
arithmetic
operators
+
,
−
,
∗
,
and
÷
.
Ordinary
dif
ferential
equation
ha
v
e
dif
ferent
types
of
ODE
problems.
Here,
we
consider
it
in
the
algebraic
w
ay
.
W
e
use
mapping
to
represent
the
characteristic
transformation
processing
(see
on
Figure
5).
Figure
5.
The
diagram
of
measurement
set
map
The
authors
denote
Algebras
by
C
the
set
of
comple
x
numbers.
A
C
-algebra
A
is
a
C-v
ector
space
that
is
also
a
ring
[11],
[12].
The
y
also
proposed
that
lter
spaces
is
algebras,
lter
as
metrics.
F
or
e
xample,
if
module
M
is
of
is
of
dimension
n
with
basis
b
=
(
b
0
,
...,
b
n
−
1
)
and
s
∈
C
n
,
then:
ϕ
(
s
)
=
s
=
n
−
1
X
i
=0
s
i
b
i
(1)
and
its
z-transform,
fourier
transform.
Some
realization
of
the
abstract
space
model
is
sho
wn
in
T
able
3.
T
able
3.
Realization
of
abstract
model
Concept
Abstract
Realized
Shift
opetator
q
T
1
(
x
)
=
x
Space
mark
t
n
C
n
K-fold
shift
operator
q
k
=
T
k
(
q
)
T
k
(
x
)
Signal
P
s
n
t
n
P
s
n
C
n
(
x
)
Filter
P
h
k
T
k
(
q
)
P
h
k
T
k
(
x
)
Assumption:
the
static
and
dynamic
feature
of
(sensor
or
control)
system,
are
often
also
d
e
ned
by
error
and
other
parameters
partial
dif
ferential
equation.
And
in
ph
ysics
w
orld,
the
enti
ty
state
is
reasonable
to
be
represented.
So,
no
r
mally
it
is
represented
by
eld,
ring,
group,
or
other
kinds
of
math
domain.
It
is
reasonable
and
con
v
enience
that
som
e
measurement
quantity
or
quality
are
represented
by
group.
An
y
this
may
be
their
inner
comparable
properties
of
measurement.
Lie
group
analysis
(symmetry
anal
ysis)
nds
point
transformations
which
map
a
gi
v
en
dif
ferential
equation
to
itself
[15].
Exce
pt
SO(2),
the
Af
f(2)
and
SO(3)
are
also
often
used.
SO(3)
is
the
group
of
rotations
in
3D
space,
represented
by
3
orthogonal
matrices
with
unit
deter
-
minant.
It
has
three
de
grees
of
freedom:
one
for
each
dif
ferential
rotation
axis.
The
in
v
erse
is
gi
v
en
by
the
transpose:
R
∈
S
O
(3)
⊂
R
3
×
3
Indonesian
J
Elec
Eng
&
Comp
Sci,
V
ol.
25,
No.
3,
March
2022:
1297–1307
Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian
J
Elec
Eng
&
Comp
Sci
ISSN:
2502-4752
❒
1303
R
−
1
=
R
T
det
(
R
)
=
1
Af
f(2)
is
the
group
of
af
ne
transformations
on
the
2D
plane.
It
has
si
x
de
grees
of
freedom:
tw
o
for
translation,
one
for
rotation,
one
for
scale,
one
for
stretch
and
one
for
shear
.
Subgroups
include
Sim(2).
After
the
discussion
abo
v
e,
there
are
some
methods
to
represent
the
measurement
set
mapping,
the
y
are
sho
wn
in
Figure
5.
The
processing
or
analysis
is
al
w
ays
undertaking
de
v
eloping
while
a
object
is
mo
ving.
The
better
method
is
the
timely
solution
and
wi
thin
the
timeout
by
some
distrib
ution
or
human
being
action.
And
also,
for
the
mo
ving
properties
of
objects,
the
best
solution
is
v
arying
with
time
[16].
The
representation
of
parameters
measurable
is
sho
wn
in
T
able
4.
T
able
4.
P
arameters
measurable
representation
General
operation
methods
Lie
Group
→
the
normalized
measurement
set
Measurement
set
and
its
map
→
nite
elds
Measurable
→
mo
v
ement
or
error
e
xist
rule
Modern
algebra
←
the
structure
of
measurement
set
3.2.
J
oint
space
mapping
Joint
space
is
calculated
by
linear
interpolation
of
each
joint.
F
or
attitude,
it
is
possible
to
nd
the
rotation
axis
between
the
tw
o
attitudes,
and
then
interpolate
t
he
rotation
angle.
Linear
interpolation
of
Lie
group:
A
α
=
A
1
⊕
α
⊙
((
⊖
A
1
)
⊕
A
2
)
,
0
≤
α
≤
1
.
In
Joint
space:
q
α
=
q
1
⊕
α
⊙
((
⊖
q
1
)
⊕
q
2
)
(2)
q
α
=
q
1
+
α
((
−
q
1
)
⊕
q
2
)
(3)
q
α
=
q
1
+
α
(
q
2
)
−
q
1
)
,
0
≤
α
≤
1
(4)
for
attitude:
R
α
=
R
1
⊕
α
⊙
((
⊖
R
1
)
⊕
R
2
)
(5)
R
α
=
R
1
+
α
((
−
R
1
)
⊕
R
2
)
(6)
R
α
=
R
1
·
(
R
1
)
−
1
·
R
2
)
α
,
0
≤
α
≤
1
(7)
4.
EXPERIMENTS
AND
SIMULA
TION
4.1.
Action
r
ecognition
using
lie
netw
ork
based
on
sk
eleton
data
set
F
or
sk
eleton
data
has
follo
wing
feature
[16]:
each
input
is
an
element
on
the
lie
group.
The
sk
eletal
data
can
be
e
xhibitted
as
fully
connected
con
v
olution-lik
e
layers
and
pooling
layers,
it
has
a
deep
netw
ork
architecture
of
the
Lie
group.
The
train
processing
is
sho
wn
in
Figure
6.
The
measur
ement
set
r
epr
esentation
of
the
body
postur
e
based
on
gr
oup
theory
...
(Changjian
Deng)
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1304
❒
ISSN:
2502-4752
Figure
6.
The
training
processing
of
lie
netw
ork
4.2.
Body
postur
e
r
ecognition
by
MPU6050
It
is
the
direct
angle
dat
a
obtained
by
the
data
fusion
algorithm
between
gyroscope
and
accel
eration
sensor
.
The
deplo
ying
place
MPU6050
sensor
is
sho
wn
in
Figure
7.
Lik
e
sk
eleton
data
and
lie
group
feature,
dif
ferent
body
postures
are
sho
wn
in
Figures
8-11.
It
is
ob
vious
that
the
relationship
of
Euler
angle
fol
lo
ws
the
theory
of
rigid
body
mechanics.
And
it
is
foundation
of
recognition
of
body
posture.
Figure
7.
The
MPU6050
sensor
deplo
ying
place
on
body
Indonesian
J
Elec
Eng
&
Comp
Sci,
V
ol.
25,
No.
3,
March
2022:
1297–1307
Evaluation Warning : The document was created with Spire.PDF for Python.
Indonesian
J
Elec
Eng
&
Comp
Sci
ISSN:
2502-4752
❒
1305
Figure
8.
The
body
posture
no.
1
Figure
9.
The
body
posture
no.
2
Figure
10.
The
body
posture
no.
3
The
measur
ement
set
r
epr
esentation
of
the
body
postur
e
based
on
gr
oup
theory
...
(Changjian
Deng)
Evaluation Warning : The document was created with Spire.PDF for Python.
1306
❒
ISSN:
2502-4752
Figure
11.
The
body
posture
no.
4
5.
FUTURE
STUD
Y
There
are
some
paper
[17]-[21]
that
study
the
coding-encoding
technology
,
in
the
future
research
of
the
manuscript
we
should
consider
the
method
based
on
concepts
of
group.
In
articles
[22]-[25]
research
the
mathematical
e
xpression
of
symmetry
in
ph
ysics
using
group
theory
,
in
the
future
research
of
the
manuscript
we
should
studies
symmetry
in
ph
ysics
more
deeply
.
6.
CONCLUSION
The
paper
proposes
a
body
posture
measurement
set
representation
method
based
on
concepts
of
group,
elds,
ring.
It
attempts
to
e
xplore
the
intrinsic
relationship
among
numerous
dif
ferent
measurement
set.
Meanwhile,
the
paper
maps
body
posture
data
into
image.
The
contrib
utions
of
paper
are:
1)
it
proposes
image
representation
method;
2)
it
presents
body
posture
representation
and
analysis
method
based
on
lie
group.
A
CKNO
WLEDGMENT
The
paper
is
supported
by
2019R
YJ03
Open
fund
project
of
Sichuan
K
e
y
Laboratory
of
articial
intelligence.
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