TELKOM
NIKA
, Vol.11, No
.3, March 2
0
1
3
, pp. 1148 ~ 1156
ISSN: 2302-4
046
1148
Re
cei
v
ed O
c
t
ober 1, 20
12;
Revi
se
d Ja
n
uary 3, 2013;
Acce
pt
ed Jan
uary 15, 201
3
Modified MVR-CORDIC Algorithm and it’s Application in
Attitude Measur
e
ment
Jiang Hai
t
ao
, Shang Xiaoxing
Dep
a
rtment of Ph
y
s
ics an
d El
ectronics e
ngi
n
eeri
ng, Jiaoz
uo
T
eachers Co
ll
ege, Jia
o
zu
o, Chin
a,
Shan Ya
ng L
u
998, Sha
n
yan
g
District, Jiaoz
uo Cit
y, Hen
a
n
Province, Ch
in
a
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: jzchao
na
n@
ya
ho
o.com.cn
A
b
st
r
a
ct
Aiming
at attitude meas
ure
m
e
n
t w
i
th 51 MCU,
T
he spee
d and effici
ency
of basic MVR-
CORDI
C
(Modifie
d
Vector Rotation
al-
C
oor
din
a
te Ro
tation Dig
ital
C
o
mputer) are l
o
w
e
r,
w
e
lists
the steps used
to
me
asur
e the head
ing i
n
rotate mo
de. Cons
ider
ed
the feat
ures of the
51
-MCU and the
MVR-CORDI
C
alg
o
rith
m, some new
scheme
s
about
iterati
o
n nu
mb
er and
search
ing a
l
go
rithm ar
e pres
ented to i
m
pr
o
v
e
the spe
ed. this
mo
difie
d
al
gor
ithm
is tested
and c
o
m
par
ed
in 51-MC
U, the exp
e
ri
me
nt result sh
ow
s that
ada
ptive iterati
on mak
e
s
iter
a
t
ion
ti
mes t
o
6
7
perc
ents of t
he g
e
n
e
ral
nu
mb
ers a
nd th
e
mo
difi
ed se
arc
h
in
g
alg
o
rith
m re
du
ces 57 p
e
rce
n
ts of t
he nor
ma
l iteratio
n ti
me.
T
he exper
i
m
e
n
tal resu
lts sh
ow
that the Angl
e
indic
a
tor system
has high resolution;
accur
a
cy of pitch A
ngle within the
±0.4°; the acc
u
racy of roll wi
thin
the ±0.4° when pitch angle is
not more than
80°, which meet the m
e
asuring system
acc
u
racy.
Ke
y
w
ords
: attitude
meas
ure
m
e
n
t, MVR-
CORDIC, adapti
v
e iteratio
n
Copy
right
©
2013 Un
ive
r
sita
s Ah
mad
Dah
l
an
. All rig
h
t
s r
ese
rved
.
1. Introduc
tion
The te
chn
o
lo
gy of slantin
g
mea
s
ure is t
o
det
e
r
mine t
he tilt and ten
den
cy of the
obje
c
t in
the spa
c
e. Til
t
-sen
so
r is a
device
whi
c
h
is used
to me
asu
r
e the in
cl
ine of the obj
ects
com
pared
with the h
o
ri
zontal. This te
chn
o
logy ma
y be appli
ed i
n
many a
s
pe
cts, such a
s
i
nertia m
e
a
s
u
r
ing
s
y
s
t
em of s
p
ac
ec
raft, determining extens
ional of
robot manipulator,
inc
line measuring of vehicl
e
and hull, jud
g
i
ng of rock te
nden
cy, contrail detectin
g
of broa
ch, etc.
Gene
rally the acceleratio
n
sen
s
or an
d signal co
nditioning
circuit integrat
ed with
comp
one
nts
calle
d accel
e
romete
r. Acceleratio
n
sen
s
or i
s
a
kin
d
of Angle
sen
s
o
r
, whi
c
h
measures th
e value of acceleration o
f
sports o
r
g
r
avity and converts to el
ectri
c
sig
nal
s o
f
sen
s
o
r
s. Th
e purp
o
se of accele
ration m
e
asu
r
em
ent
is
to get the attitude Angle of
the obje
c
t.
In the Angle
sen
s
o
r
sign
al
proce
s
sing, functi
o
nal ope
ration algo
rith
m to the precision of
the output value plays a de
cisive role.
If
the gene
ral look-up table
or interp
olatio
n calculation
is
employed, th
e accu
ra
cy a
nd computin
g
spe
ed
sh
o
u
l
d
give way to each othe
r i
n
the Arctan
and
open radi
cal
sign fun
c
tion
operation, wit
h
more
h
a
rd
ware re
so
urces be o
c
cupi
ed [1]. There
are
a variety of
Arctan fun
c
ti
on cal
c
ul
ation met
hod
s, like loo
k
-up table metho
d
[2], polynomial
approximatio
n method [3], rational approx
imatio
n [4], CORDI
C algorith
m
, etc.
the method o
f
Loo
k-u
p
tabl
e is speedy,
simple a
n
d
easy-to
-
do,
but take u
p
too big m
e
mory capa
city;
Polynomial approximatio
n algorithm
can ac
hie
v
e higher pre
c
isi
on, but the general
approximatio
n orde
r of polynomial num
ber is high
er, large am
ount
s of calcul
atio
n, and con
s
u
m
e
a large amo
unt of hardware re
sou
r
ce
s; Rati
onal a
pproxim
ation,
compa
r
ed with polynom
ial
approximatio
n method, i
s
more li
kely
to achi
eve
highe
r preci
s
ion, but a la
rge a
m
ount
of
comp
uting time is con
s
u
m
ed in block operation .T
he Coo
r
din
a
ted Rotation
Digital Com
p
uter
(CORDIC) al
gorithm i
s
well-kno
w
n ite
r
ative te
ch
ni
que to pe
rfo
r
m vario
u
s
basi
c
a
r
ithm
etic
operation
s
[1]-[3]. The algorithm is very attracti
ve for hard
w
a
r
e imp
l
ementation b
e
ca
use it uses
only element
ary shift-a
n
d
-
add ste
p
s to
perfo
rm vecto
r
rotation in
a
two-dim
e
n
s
i
onal (2
-D) pla
ne.
Hen
c
e, the CORDIC algo
ri
thm can be applied to
many DSP application
s
wh
ere rotation-ba
sed
arithmeti
c
functions are heavily ut
ilized, such as linear sy
stem
solv
er [4], [5], dig
i
tal lattice filter
[6], [7], singul
ar value p
r
obl
ems [8].
In this paper, we propo
se a modifie
d
ve
ctor rot
a
tional CO
RDIC (MVR-CORDIC)
algorith
m
[9], It is very suitable for appli
c
atio
n
s
that use the CORDIC algorithm in only forward
rotation mod
e
(also kno
w
n as vecto
r
rota
tion mode), i.e, the rotation angles a
r
e fixed and kno
w
n
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 2302-4
046
TELKOM
NIKA
Vol. 11, No
. 3, March 20
13 : 1148 – 1
156
1149
in advan
ce, such a
s
digital
lattice filter [6], [7
],
[10] and discrete li
near tr
an
sformation [11], [12].
The majo
r fe
ature of the
aforem
ention
ed appli
c
atio
ns i
s
that the dire
ction
a
l seq
uen
ce,
()
i
,
whi
c
h co
ntrol
s
the rotation
direction of each
element
ary angle in the micro rota
tion phase, can
be co
mputed
in advan
ce. By reformatting
and se
ar
chin
g for ne
w se
q
uen
ce
s, we can red
u
ce th
e
iteration n
u
m
ber
signifi
ca
ntly, while n
o
t increa
se t
he qua
ntizati
on noi
se lev
e
l. This
can
be
achi
eved by modifying the basic mi
cro
rotati
on procedure of con
v
enti
onal CO
RDI
C algorith
m
.
Then, we
can
improve the
spe
ed pe
rformanc
e of the conve
n
tional
CO
RDI
C algo
rithm.
2. The Propo
sed Algori
t
h
m
2.1 MVR - CORDIC princ
i
ple
CO
RDI
C alg
o
rithm is a
ki
nd of vector
rotation
for th
e cal
c
ulatio
n of iterative algorithm.
Its basic ide
a
is to decompose a pre-ro
tated spe
c
if
ic Angle, with linear combi
n
a
t
ion of a set
of
pre
s
crib
ed ba
sic Angl
e, na
mely, the multiple basi
c
An
gle rotation .i.e:
X
Z
Y
o
(1)
(N =W=ba
s
i
c
Angle element quantity
;
}
1
,
1
{
)
(
i
, the
firs
t ti
me rotating direc
t
ion;
)
2
arctan(
)
(
i
i
,
the corre
sp
ondin
g
basi
c
Angle elemen
ts;
,
the after-iterative resi
d
ual error of
the angle
)
.
In contr
a
st wi
th the basi
c
CO
RDI
C alg
o
rithm, MVR
- CO
R
D
IC c
h
ara
c
teri
ze
s in
: (1)
ski
ppin
g
so
m
e
micro-
rota
tion Angle; (2) re
peat
in
g some micro-rotation
Angle
;
(3)
redu
cin
g
iteration
s
, in
orde
r to g
r
eatly improv
e the it
erati
v
e spe
ed a
nd iteratio
n
pre
c
isi
on. Th
us,
decompo
sitio
n
expre
ssi
on
of MVR - CO
RDI
C ha
s be
en ch
ang
ed from (1) into
(2
):
1
0
))
(
(
)
(
m
R
i
i
S
i
(2)
(
m
R
, iteration
s
, generally
W
R
m
;
}
1
,
0
,
1
{
)
(
i
,
the i-th rotating direction;
()
(
(
))
arct
an
(
2
)
Si
Si
, the co
rre
sp
ondin
g
ba
sic
Angle ele
m
e
n
ts
when th
e
i-th micro
-rot
a
ted, in
whi
c
h
}
1
,
2
,
1
,
0
{
)
(
N
i
S
.)
This pa
per u
s
e MVR - CO
RDI
C algo
rith
m vector mo
del [13], in which
()
i
is a sym
bol
function, its value dete
r
min
ed by
)
(
i
Y
. Iterative equatio
ns
can be written as follo
ws:
)]
(
[
)
(
)
(
)
1
(
2
)
(
)
(
)
(
)
1
(
2
)
(
)
(
)
(
)
1
(
)
(
)
(
i
S
i
i
Z
i
Z
i
X
i
i
Y
i
Y
i
Y
i
i
X
i
X
i
s
i
s
(3)
(0
)
,
(0
)
XY
(X
0
, Y
0
here
a
fter)
Co
rrespondi
ng re
q
uest Angle
co
sine a
nd
sinu
soi
dal
comp
one
nt,
0
)
0
(
Z
;
In the beginni
ng of iteration ,select
the approp
riate S(i), to make Y(i+1
)
minimum; take
()
,
(
)
iS
i
into the
type X(i+1) o
f
iterative equation
s
to
get corre
s
po
nding
X(i+1
)
; loo
k
-up table to g
e
t
((
)
)
Si
, and thus
get Z(i
+
1),
s
o
far co
mplete
a iterative; After
m
R
iteration, re
su
lts of output as follow:
)
/
arctan(
)
1
(
0
)
1
(
)
1
(
0
0
2
0
2
0
X
Y
i
Z
i
Y
Y
X
P
i
X
(
1
0
)
(
2
2
1
m
R
i
i
s
P
)
(4)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Modified MV
R-CO
RDI
C Algorithm
and i
t
’
s
Applicat
ion in Attitude Measurem
ent (Jian
g
Haita
o
)
1150
From equ
atio
ns (4
), we ca
n see that, due to
the existence of the s
c
alin
g-fa
ctor P and its
uncertainty, so scale-fa
cto
r
cal
c
ulatio
n and vect
o
r
module comp
e
n
satio
n
are n
eede
d, that is,
P
K
/
1
,we ca
n write
as.
)
1
(
)
1
(
'
i
X
K
i
X
(5)
In the actu
al
use, vari
ou
s i
m
provem
ent
of
MVR--CO
RDI
C alg
o
rith
m is ma
de to
improve
the efficien
cy
and
spe
ed o
f
the algo
rith
m; For ex
a
m
ple, acco
rdin
g to the
h
a
rd
ware stru
cture
of
FPGA, propo
sed the use of check RO
M table in
stead of the original algo
rith
m scale
-fa
c
tor
cal
c
ulation method [13].In
this
paper, MVR - CORDIC algorit
hm, whi
c
h had been modified and
improve
d
, is applie
d to cal
c
ulate a
ngle
based on 5
1
-MCU.
2.2 Algorith
m
Impro
v
ement
In the realizat
ion of algorith
m
, the basic
MV
R - CO
RDIC pre-rotate
d Angle is mo
dified to
find out the
effective bits, calcul
ation formul
a of basic Angle an
d the number of element,
and
employed ad
aptive iteration times and the met
hod of improved-rotation sequ
ence prop
osed.
The improved
algorithm flo
w
ch
art a
s
Figure.1:
Figure 1. MVR-CO
RDI
C flow chart
First of all, in orde
r to make the algorit
h
m
conve
r
gen
ce in the wh
o
l
e plane, and
achi
eve
highe
r a
c
curacy, the p
r
e
-
rotation
of th
e initial
Angl
e
nee
ded
ca
rrying out, i.e. The step
2 of
figure 1, and
to
get
(1
)
[
0
,
/
4
)
Zi
,Corre
spo
ndin
g
ly, a
t
the same time,
0
X
,
0
Y
should
be
pretreated to
make
)
4
/
,
0
[
)
/
arctan(
0
0
X
Y
.
This p
ape
r pretreate
d
0
X
,
0
Y
to
mak
e
0
0
0
0
,
0
,
0
Y
X
X
Y
, and cod
ed the
Re
sult (as ta
ble 1). After the iteration, the iter
ation result
s are rev
e
rsely pro
c
e
s
sed to get the
real Angl
e value acco
rdin
g to the code v
a
lue.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 2302-4
046
TELKOM
NIKA
Vol. 11, No
. 3, March 20
13 : 1148 – 1
156
1151
Table 1. Rota
tion vector ini
t
ial value pret
reatme
nt and
codin
g
Coding
Initial value
Pre-rot
a
ti
on
Result
0 [0,
π
/4)
'
'
0
0
Y
X
=
0
0
Y
X
0
1
[
π
/4,
π
/2)
'
'
0
0
Y
X
=
0
0
X
Y
π
/2-
θ
2
[
π
/2,3
π
/4)
'
'
0
0
Y
X
=
0
0
X
Y
θ
-
π
/2
3 [3
π
/4,
π
)
'
'
0
0
Y
X
=
0
0
Y
X
π
-
θ
Secon
d
ly, in the improve
d
algorith
m
, data e
ffective length is deter
mined by accura
cy of
angle
whi
c
h is determine
d by the syste
m
requi
rem
e
nts. Cal
c
ulati
ng formul
a as follows:
)
2
arctan(
)
arctan(
N
x
=>
N
2
1
1
'
(6)
In whic
h
N
x
2
, N=data
sig
n
ificant bit len
g
th. Furthe
rmore, the
n
u
mbe
r
of the
element
s of b
a
si
c Angle
se
t of im
proved algorith
m
is d
e
termin
ed
by
effective bit length. Fo
rmu
l
a
is
as
follows
:
N
m
m
2
1
)
2
arctan(
)
(
(7)
In whi
c
h m
= the minimum
numbe
r of el
ements
withi
n
the ba
sic A
ngle
set. In the ba
si
c
MVR-
CO
RDI
C algo
rithm,
m
R
,
a con
s
tant, see type (2), is defin
e
d
the minimum of iteration
s
to
satisfy the p
e
r
forma
n
ce of
algorith
m
. in
most
ca
se
s, the a
c
tual ite
r
ations
are
less than
m
R
in
the
actual cal
c
ula
t
ion.
Adaptive iteration meth
o
d
di
d
not p
r
ovide
co
nst
ant iteratio
n
s
, but a
u
to
matically
determi
ned b
y
iterative proce
s
s itself.
Whe
n
t
he al
gorithm m
e
t the pre
c
i
s
io
n req
u
ire
m
e
n
t,
namely, end
iteration p
r
o
c
ess, to avoid
unne
ce
ssary
iterative pro
c
e
ss
and th
u
s
imp
r
ove th
e
efficien
cy. In
the applicatio
ns,
'
mm
RR
is given to ensure the
pro
c
e
ss
of al
gorithm in a controlle
d
way
.
The improve
d
algorithm p
r
ovided the
way to
search the rotary se
quen
ce. The
step 5 of
figure.1, the
right
)
(
i
and
)
(
i
S
are
need
ed to mi
nimize
(1
)
Yi
. According to th
e p
r
i
n
cipl
e, the
value of
)
(
i
can
be dire
ctly determine
d by
)
(
i
Y
,
So the step 5 is main to determin
e
the value
of
)
(
i
S
.So, Quickl
y finding suit
able
)
(
i
S
is an effective way to
improve the
spee
d of iteration
.
From type (2), we can
see
1
,
,
2
,
1
,
0
)
(
N
i
S
is an incre
a
si
ng se
q
uen
ce;
From
t
he
iterative equa
tions (3
) an
d iterat
ive pro
c
e
ss, we ca
n find out:
)
(
)
1
(
i
X
i
X
;
)
(
)
1
(
i
Y
i
Y
;
=>
)
(
)
1
(
i
S
i
S
(8)
Improved se
arch method
cha
r
a
c
teri
ze
d
in that, using the
)
(
i
S
, which is saved by the
step 6, determine the ran
ge of
S
(
i
+1)
on step 5 of the next iterat
ion pro
c
e
ss
; so, searchi
n
g
rang
e of
)
(
i
S
would decrea
s
e
with i increa
sing, wh
i
c
h speed
ed up the searchi
ng spe
ed and
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Modified MV
R-CO
RDI
C Algorithm
and i
t
’
s
Applicat
ion in Attitude Measurem
ent (Jian
g
Haita
o
)
1152
cos
s
i
n
0
sin
c
os
0
00
1
R
sho
r
ten the
step 5 o
p
e
r
a
t
ion time ; time of si
n
g
le
iteration al
so red
u
ced
correspon
dingl
y,
improvin
g the
spee
d of execution.
The
scale-fa
ctor is
pro
d
u
c
ed du
ring t
h
e
iterati
on,
se
e type (4
). T
he Scale-fa
ct
or i
s
al
so
cha
ngin
g
because of rotary sequen
ce a
nd iterati
on times of unce
r
tainty. The algorithm alwa
ys
dire
ct cal
c
ula
t
e scal
e
fact
or in the iterati
on pro
c
e
s
s, and then
prod
uct the
previou
s
st
orage
factor a
nd sto
r
age. see ste
p
(7) in Fig
u
re1.
3. Rese
arch
Metho
d
3.1. Angle M
easur
e
ment
Principle
Accel
e
ration
sen
s
o
r
can b
e
use
d
to determin
e
ch
ang
es o
r
co
nsta
n
t
accel
e
ration
. On the
earth any po
sition obj
ect
s
are subje
c
t to the e
ffect of gravity and prod
uce a accele
ration
and
accele
ration
of gravity is a spe
c
ial case of con
s
tant a
c
celeration.
Acco
rdi
ng to the rigid bo
d
y
dynamics p
r
inci
pl
e, it is kno
w
n that th
e Angle between the
referen
c
e co
ordin
a
te syst
em bound to
gether
with moving obje
c
t and fixed referen
c
e
coo
r
di
nate
system
can b
e
calle
d the inclin
ed st
ate
(nam
ely attitude).see Fig
u
re 2.
Figure 2. Coo
r
dinate
syste
m
At this time, yaw angl
e, pitch angl
e
and ro
ll an
gl
e in the spa
c
e can b
e
expressed a
s
follo
ws:
Ya
w
angle
(
):
angle
betwe
en OX shaft's the proj
ectio
n
line in h
o
ri
zontal plan
e (O X
d
Y
d
)
and
OX
d
s
haft.
Pitch angle (
):
an
gle bet
ween OX shaft and the ho
rizontal plan
e (OX
d
Y
d
)
Roll angle (
):
angl
e bet
ween the
sym
m
etry plan
e
of t
he obje
c
t
and the ve
rti
c
al pl
ane
whi
c
h
contai
ns X
d
shaft.
T
h
r
e
e
r
o
ta
tion
s
co
rr
es
p
ond
ing with three matrixes
,
which
can b
e
written as:
10
0
0c
o
s
s
i
n
0s
i
n
c
o
s
R
In the ide
a
l condition
s, the
Mea
s
ureme
n
t
value
of an
g
l
e se
nsor
or t
r
iaxial a
c
cele
rometer
can b
e
norma
lized a
s
:
G
R
R
R
G
m
si
n
s
i
n
c
o
s
c
o
s
[s
]
co
T
.
Whe
r
e:
[0
0
1
]
T
G
the no
rmali
z
ed exp
r
ession G
r
avit
y fiel
d (namel
y g ) in defini
t
e reference
coo
r
din
a
t
e
sy
st
em.
From G
m
, we can de
duce Pitch an
gle and roll a
ngle expressi
on as follo
ws:
22
arct
an
(
)
mx
my
m
z
G
GG
mz
my
G
G
arctan
(9)
cos
0
s
i
n
01
0
si
n
0
c
o
s
R
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ISSN: 2302-4
046
TELKOM
NIKA
Vol. 11, No
. 3, March 20
13 : 1148 – 1
156
1153
3.2. Data M
e
asureme
nt P
r
inciple
Angle se
nso
r
wa
s install
ed on a ho
rizo
ntal rotati
ng
table fre
e
of magnet
during
Experiment;
in order to
validate the
data, output
sign
al from
the sensor,
throu
gh
serial
transmissio
n, to computer was tran
smit
ted afte
r A/D sampling. the seri
al given
orders, singl
e-
chip gath
e
re
d the enco
d
e
r
and tran
smi
tted to the
c
o
mpute
r
; after that rotation level turntable
wa
s rotated. Operation ab
ove duplicate
d
, level
turntable wa
s rot
a
ted from the 80 degre
e
s
in
Angle of pitch and 280 de
gree
s in angl
e of roll, re
sp
ectively to gather a grou
p of data every 10
degree
s, an
d
then record
ed the
accel
e
rom
e
ter
out
put value
an
d the
corre
s
pondi
ng Angl
e of
pitch an
d roll;
a total of 16
x16 grou
p dat
a wa
s mea
s
u
r
ed.
Ideally, Angle of pitch a
nd roll from
the
turntabl
e sh
ould b
e
con
s
i
s
tent
with that
displ
a
yed o
n
the co
mpute,
however, d
u
e
to all ki
nd
s of error i
n
th
e actu
al me
a
s
ureme
n
t. Erro
r
equally exist
ed betwe
en
the measu
r
ement re
su
lt
s and sta
n
d
a
rd on
es. Accordi
ng to our
acq
u
isitio
n d
a
ta, corre
c
tio
n
coeffici
ent wa
s utiliz
e
d
to do the calculation, and t
hen ap
plied t
o
the
system, Angl
e of pitch and roll measu
r
ed rep
eatedl
y was comp
ared with sta
ndard turnta
ble
comp
ari
s
o
n
; the data sho
w
ed that data vibrated am
on
g [- 80°, 80°]
rang
e, with error bel
ow 0.4
°
,
whi
c
h met the requi
rem
e
n
t
of experime
n
t.
Becau
s
e
of the ci
rcuit itself and the
outsid
e
environment fa
cto
r
s, it is inevi
t
able to
commit erro
rs in angle mea
s
ureme
n
t system. Acco
rd
in
g to the nature and form of
expressio
n
o
f
the error, Errors
ca
n be di
vided into sy
stem error,
ran
dom erro
r an
d gro
s
s erro
r. The sy
stema
t
ic
error ke
eps
consta
nt
or ch
ange
s with
a certai
n
la
w, whi
c
h can be
com
pen
sate
d
and co
rrect
ed;
rand
om e
rro
r is unp
re
dict
able an
d un
correcte
d, but
statistically estimabl
e in
its implications;
gro
ss e
r
ror i
s
ca
used by
human fact
ors, with the
erro
r value
gene
rally mo
re obviou
s
a
n
d
avoided. The
chapte
r
deal
s with an
alysis of syst
em
a
t
ic error, the
error comp
en
sation a
nd its
verific
a
tion.
3.3. Error An
aly
s
is
Er
r
o
r
fa
c
t
or
s
o
f
a
c
ce
le
ra
tion
s
e
ns
or
in
th
e
field of ine
r
tial navigatio
n have bee
n studie
d
,
it is ge
nerally accepte
d
tha
t
the
error
of accele
ration
sen
s
o
r
lie
s
m
a
inly in the i
n
stallation
error,
sen
s
itivity error, cross axis
se
nsitivity, cro
ss
cou
p
ling error an
d
random e
r
ror; for the low
pre
c
isi
on of micro mech
a
n
ical accel
e
ration sen
s
o
r
, its error source was negligible in terms of
order two or more error coeffic
i
ent.
A tilt sensor
stru
cture sh
o
w
s u
s
, wh
en
the
inclinati
on se
nsor be
ing used to do static
attitude measurem
ent, with no lateral displacement
effect, the output signal of the sen
s
or i
s
not
affected by a
c
celeration in
terfere
n
ce [14]. The main sign
al error
compon
ents:
(1) Zero err
o
r
w
i
th tem
p
erat
ure
:
ze
ro error d
r
ive
n
by change
s in the temperatu
r
e
value and te
mperature
co
efficient, temperatu
r
e
ca
n
be ch
ang
ed
by instrum
e
nt calib
ration
to
redu
ce, temp
eratu
r
e coefficient effect c
an be le
ssen
ed by temperature comp
en
sation.
(2) Error
w
i
th sine functi
on:
th
e
s
i
n
g
l
e
a
x
is
a
c
c
e
le
ro
me
te
r
s
i
g
n
a
l
o
u
t
p
u
t
is
p
r
op
o
r
tional
to
sin(x)
, where x is a
singl
e axis a
c
celeromete
r tilt angle. In the mea
s
u
r
eme
n
t of large
amplitude a
n
g
le, this effect is seen a
s
a
nonline
a
r ad
ded value.
(3) Er
ror
w
i
th v
i
bration:
su
ch an error with vibration may be caused by slight vibration
of experime
n
t platform in e
x
perime
n
t en
vironme
n
t, as oppo
sed to the req
u
ired st
ationary si
gn
al.
(4) Error
w
i
t
h
v
o
ltage ch
anges
:
for b
e
st re
sult
s, the voltage sho
u
ld be mai
n
tained at
5V; if the voltage is
cha
n
g
ed, output sin
gal will ra
nge
within 2% in error.
(5) Error
w
i
t
h
installatio
n
:
du
ring in
st
allation, thre
e
sen
s
o
r
mea
s
uri
ng
shafts are not
parall
e
l to ca
rrier in lo
ngitu
dinal, tran
sve
r
se,
verti
c
al vector of three
axis, respecti
vely.
3.4. Error Co
mpensa
tion Metho
d
3.4.1. Nonlinear Error Co
mpensa
tion Metho
d
Single axis a
c
celeromete
r output sig
n
a
l
is pr
opo
rtio
nal to the si
n
e
value of in
clinatio
n
angle, theref
ore, we ado
p
t
the linearization me
thod
of sine value and accele
ration, each axle
accele
ration
comp
en
sated
.
Specific met
hod
s are a
s
follows:
(1): first calcu
l
ate corre
s
po
nding an
gle X wi
th each sine value, then determin
e
the axis
of calibratio
n
accel
e
ro
met
e
rs, after tha
t
rotate axis to angle X
in calibration,
reco
rd outp
u
t
accele
ration
value of calib
ration shaft in the Table 2.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Modified MV
R-CO
RDI
C Algorithm
and i
t
’
s
Applicat
ion in Attitude Measurem
ent (Jian
g
Haita
o
)
1154
Table 2. i-axi
s
nonli
nea
r compen
satio
n
sampli
ng dat
a
Sinx
-1
-0.9
-0.8
-0.7
-0.6
…
0.6
0.7
0.8
0.9
1
a
i
…
…
… … … … … …
…
…
…
(2): curve fitting with leas
t s
q
uare method
Curve fitting
was de
sig
n
ed to use a
simple fun
c
tion to appro
x
imate a co
mplex or
unkno
wn fun
c
tion, with fu
nction valu
e of finite sa
mp
ling point
s in an interval a
s
target [25]. Let f
(x) be the interpol
ation function, P (x)
a poly
nomial
function, when the
sum of square of
the
differen
c
e of the two functi
ons was mini
mal, we
may
call P (x) as the best sq
uare approxim
ation
of f (x) [15].
Curve fitting of least squa
res i
s
to see
k
a coefficie
n
t vector, wh
ich is a polynomial
coeffici
ent. Linear
equ
ation of each axis in the de
sig
n
is: Sin (x) = k*ai +
b, i.e. to figure out t
h
e
value k an
d b
,
where x is a
single axi
s
a
c
celeromete
r tilt angle.
3.4.2. Installation Error Compensa
tio
n
Method
Acco
rdi
ng to the angle me
asu
r
e pri
n
ci
p
a
l, we
may know that, ideally, the outp
u
t signal
of three axis
meets the eq
uation group
s in 10:
cos
cos
cos
sin
sin
r
g
Az
r
g
Ay
g
Ax
(10
)
In a no
nide
al enviro
n
ment
, due to the
su
ch in
co
nsi
s
tencie
s of th
e thre
e outp
u
t sign
al
from thre
e
axis a
c
celerometers a
n
d
pre
s
en
ce
o
f
erro
r after the input
signal bei
ng
the
system
aticall
y
magnified,
sampl
ed, a
c
celera
tion
sensor
calib
ra
tion pro
c
e
s
s, based on t
he
analysi
s
of errors, ca
n be e
quivalent to 1
2
coeffici
ents
in formula (11
)
1
1
1
12
13
21
22
2
3
2
31
32
3
3
3
xx
yy
zz
d
Ac
c
c
a
Ac
c
c
a
d
Ac
c
c
a
d
(11
)
Comp
en
satio
n
matrix coeff
i
cient
s are
stored
in a Fla
s
h in advance, powe
r
MCU, loaded
into the on
-chip RAM. Inp
u
t signal
after co
m
pen
sati
on calcul
atio
n and g
e
t co
mpone
nt sig
nal
free of error.
The calibratio
n
paramete
r
s can b
e
directly
calculate
d
base
d
on th
ree axis a
c
cel
e
ration
sen
s
o
r
me
asuring val
u
e
s
. The
corre
c
tio
n
pro
c
e
s
s is
as follo
ws: Adjust the tu
rn
table to the l
e
vel
state, and a
ngle mea
s
u
r
ement syst
e
m
of the three axis acce
leration
sen
s
or is pla
c
e
d
on a
turntable, rot
a
ting table located at a differ
ent pitch angle and ro
ll angle integrated attitude
,
recording th
ree axis
se
nsor o
u
tput an
d t
he corre
s
pondi
ng pit
c
h angl
e an
d
roll a
ngle, e
v
ery
measured val
ue of the acceleratio
n
sh
al
l be satisfie
d:
ˆ
[
s
i
n
()
,
s
i
n
(
)
*
c
o
s
()
,
c
o
s
()
*
c
o
s
(
)
]
T
am
o
f
f
i
i
i
i
i
i
KG
G
(12
)
Whe
r
e:
ˆ
[,
,
]
T
mm
x
i
m
y
i
m
z
i
G
ggg
is e
a
ch me
asure
m
ent value of accelerator o
f
three axis.
[,
,
]
T
of
f
i
ox
i
o
y
i
ozi
G
ggg
is bia
s
of accelerato
r of three axis.
Therefore, th
eoreti
c
ally, as long as me
asu
r
ing poi
nts unde
r the four
po
stures
fixed,
corre
c
tion co
efficient Ka and Goff ca
n be cal
c
ula
t
ed. In order to improve the calibrati
on
pre
c
isi
on, fitting co
rrectio
n
can b
e
achi
eved by
the use of m
u
ltiple inclin
ed po
sture
mea
s
u
r
ing
point (mo
r
e than 4).
Whe
r
e:
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 2302-4
046
TELKOM
NIKA
Vol. 11, No
. 3, March 20
13 : 1148 – 1
156
1155
33
32
31
23
22
21
13
12
11
C
C
C
C
C
C
C
C
C
Ka
3
2
1
d
d
d
G
off
The mea
s
u
r
in
g system of a
ttitude was te
sted
on the tu
rntable, the result
s as follo
ws:
Table 3. Outp
ut roll angle
γ
(
θ
= 0
°
)
Table 4. Outp
ut roll angle
γ
(
θ
= 1
5
° )
code Theoretical
γ
Output
γ
Error
code Theoretical
γ
Output
γ
Error
1 0°
-0.13°
0.13°
1 0°
0.14°
0.14°
2 10°
10.22°
0.22°
2 10°
9.89°
0.11°
3 20°
19.88°
0.12°
3 20°
19.79°
0.21°
4 30°
29.79°
0.21°
4 30°
30.38°
0.38°
5 40°
40.32°
0.32°
5 40°
40.29°
0.29°
Table 5. Out
put Pitch angl
e
θ
(
γ
= 0° )
Table6. Outp
ut Pitch angle
θ
(
γ
= 1
5
° )
code Theoretical
θ
Output
θ
Error
code Theoretical
θ
Output
θ
Error
1 0°
-0.31°
0.31°
1 0°
0.21°
0.21°
2 10°
10.19°
0.19°
2 10°
9.73°
0.27°
3 20°
19.76°
0.24°
3 20°
19.78°
0.22°
4 30°
29.73°
0.27°
4 30°
30.29°
0.29°
5 40°
39.64°
0.36°
5 40°
39.73°
0.27°
4. Conclusio
n
This pap
er p
u
t forward the method of adapt
ive itera
t
ion times and improved rotation
seq
uen
ce
se
arch meth
od
by employing
the im
pr
ov
e
d
MVR -
CO
RDI
C alg
o
rith
m in 51 M
C
U
.
The expe
rim
ental re
sults
sho
w
that the Angle indica
tor system
h
a
s high
re
sol
u
tion; accura
cy
of pitch Angle within the
±0.4°; the accuracy
of roll within the ±
0
.4° when pit
c
h angle is n
o
t
more than 80
°, which mee
t
the measuri
ng syst
em accuracy. The experim
ental results sh
o
w
the algorithm
efficiency was greatly improved
.But more furth
e
r rese
arch were nee
ded t
o
optimize its th
eory and a
ppl
ication.
Referen
ces
[1]
JE Volder. T
he CORDIC trigonometric comp
uting tech
niq
u
e
.
IRE T
r
ans.
Electron. Co
mputers
. 195
9;
C-8: 330
–3
34.
[2]
JS
Walther.
A unifi
ed al
gor
ith
m
for ele
m
enta
r
y functions
. in
Sprin
g
Joint C
o
mp. Conf. 19
71; 379
–3
85.
[3]
YH Hu, COR
D
IC-bas
ed V
L
SI architectur
e
s for di
gital
sig
nal
process
i
n
g
.
IEEE Signal Pr
oc
essing Mag
.
199
2: 16–
35.
[4]
K Jainan
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