TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol.12, No.4, April 201
4, pp. 3078 ~ 3
0
8
4
DOI: http://dx.doi.org/10.11591/telkomni
ka.v12i4.4837
3078
Re
cei
v
ed Se
ptem
ber 3, 2013; Re
vi
sed
No
vem
ber 1
1
,
2013; Accep
t
ed De
cem
b
e
r
14, 2013
The Approach of Error Calibration for Three-axis
Magnetic Heading Sensor
Xiao Miaoxin
Dep
a
rtment of Electrical E
ngi
neer
ing,
Xi
n
X
ia
ng Un
iversit
y
,
East Jin Sui street, Xi
n
X
ia
ng c
i
t
y
, HeN
an pr
o
v
ince, Ch
ina
email: xia
o
m
i
ao
xi
n7
4@1
63.
com
A
b
st
r
a
ct
T
he accuracy of mag
netic he
adi
ng sens
or is reduc
ed by th
e imp
a
ct of ma
nufacturi
ng tec
hno
log
y
and
loca
l
ma
g
netic i
n
terfere
n
c
es. T
he sin
g
u
larity
of cons
t
r
aint
matrix i
n
traditio
nal c
a
li
bratio
n al
gor
ith
m
lea
d
s to unstable resu
lts. T
h
erefor
e, an i
m
prove
d
least-s
quar
e elli
psoi
d
fitting method
is propos
ed in this
pap
er
. T
he err
o
r source a
nd t
he dev
iati
on
mathe
m
atic
al
mo
del ar
e intro
d
u
c
ed. On the ba
sis of the ana
ly
sis
on sin
gul
arity of the constr
ai
nts matrix, par
ameter extracti
on by t
he i
m
pr
oved l
east-squ
a
re ell
i
ps
oid fitti
n
g
meth
od is giv
e
n. T
he new
method succ
essfully overc
o
mes
the instabil
i
ty
of the tradition
al alg
o
rith
m, a
n
d
reduc
es the c
o
mp
utatio
n lo
ad
. Simu
lati
on a
nd ex
peri
m
ent
al
res
u
lts sho
w
that this met
hod
is effectiv
e i
n
calibr
a
tin
g
ma
gnetic hea
di
ng
senso
r
.
T
he hea
din
g
pr
ecis
ion
of the sen
s
or acq
u
ire
d
a
fter calibr
a
tion
is
better than 0.4
2
º.
Ke
y
w
ords
: magn
etic he
adi
n
g
sensor, error
calibr
a
tio
n
, elli
psoi
d fitting, singu
larity
Copy
right
©
2014 In
stitu
t
e o
f
Ad
van
ced
En
g
i
n
eerin
g and
Scien
ce. All
rig
h
t
s reser
ve
d
.
1. Introduc
tion
Magneti
c
h
e
a
d
ing
sen
s
o
r
s
throug
h to th
e ea
rth's mag
netic field
me
asu
r
em
ent, in
dicate
s
the course,
u
n
mann
ed
aerial vehicl
e i
s
one
of
the i
m
porta
nt sen
s
or on
the u
n
mann
ed
aerial
vehicle. But b
e
ca
use of the
influen
ce of t
he mag
netic
material
on b
oard
and
the
limitation of the
sen
s
o
r
pre
p
a
r
ation, en
cap
s
ulatio
n
p
r
o
c
ess,
di
re
ct
measurement
of
sen
s
o
r
d
a
ta contain
s
a
variety of error, e
r
ror
co
mpen
sation
sho
u
ld
b
e
carri
ed o
u
t b
e
fore
use. At pre
s
ent m
a
in
compensation methods of
ellipse
fitting, kalman filter and neural
network, elli
psoid fitting, etc.
The least-squares ellipse fitting me
tho
d
[1] is simple, but to the
compensation effect of th
ree
axis sen
s
or i
s
limited; Kal
m
an filter [2,
3], t
he neu
ral
netwo
rk ne
e
d
s hi
ghe
r p
r
e
c
isi
on referen
c
e
datum; Ellipsoid fitting me
thod ba
se
d
on iterative
method [4
-5]
are
su
scepti
b
le to the ini
t
ial
estimate a
nd
the noi
se infl
uen
ce a
nd
sp
read, a
nd
la
rge amo
unt of
cal
c
ulatio
n; Traditio
nal le
ast
squ
a
re
s fittin
g
ellip
soid
m
e
thod to
a
c
hi
eve hig
h
p
r
e
c
isi
on
com
p
e
n
satio
n
[6], b
u
t the al
gorit
hm
exist due to the insta
b
ility probl
em
s ca
u
s
ed by
the co
nstrai
nt matri
x
is singul
ar.
In order to solve this pro
b
lem, this p
aper p
u
ts forward the improved le
ast-squ
a
re
s
ellipsoid fitting method. Th
e method is
based on the
assumption
ellipsoid, erro
r com
pen
sati
on
coeffici
ent was
cal
c
ulate
d
by the lea
s
t
squ
a
re
meth
od, the
con
s
t
r
aint h
a
s
bee
n solve
d
by t
he
matrix deco
m
positio
n of matrix singul
ari
t
y problem
, to overcome the instability of the algorit
hm,
at the
sam
e
time
red
u
ce
d the
amo
u
n
t
of calculati
on. And
thro
ugh
software
sim
u
lation
a
n
d
experim
ent verify the effectiv
eness of the algorith
m
[8-10].
2. The Error Analy
s
is and Compen
sa
tion Algorith
m
For three axi
a
l magn
etic h
eadin
g
se
nso
r
, and it
s in
h
e
rent e
r
ror i
s
mainly cha
r
a
c
teri
zed
by zero error,
erro
r sen
s
itivity,
orthogona
l erro
r, etc. Assume that
th
e actual o
u
tp
ut of the sen
s
or
is
s
h
. No erro
r e
x
ists for th
e i
deal o
u
tput is
t
h
.
s
t
hh
, and its m
a
thematical
m
odel i
s
exp
r
e
s
sed
as availa
ble a
s
:
s
dp
t
e
e
t
e
hK
K
h
B
K
h
B
(1)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
The Approa
ch of Error
Cal
i
bration fo
r Three
-
a
x
i
s
Ma
gnetic
Hea
d
in
g Senso
r
(Xia
o Miaoxi
n
)
3079
Erro
r matrix
d
K
is a third order dia
gon
al
matrix,
which stand
s for the sen
s
itivity o
f
v
a
riou
s sh
af
t
sen
s
or.
p
K
stan
ds for no
n o
r
thogo
nality betwe
en
the axis of sen
s
ors a
nd soft
magneti
c
mat
e
rial pa
rt. Then a sen
s
o
r
prop
er refere
nce
coo
r
din
a
te system can
be establi
s
h
ed.
p
K
can b
e
pre
s
e
n
ted by third orde
r diag
on
al matrix.
e
B
stands for
sen
s
o
r
's
zero erro
r and ha
rd
magneti
c
ma
terials. Erro
r comp
en
sati
on of se
ns
ors equ
al to d
e
termin
e the
error
coefficient
matrix
e
K
and
e
B
. By the known
actual outp
u
t
s
h
, To solve the ideal outp
u
t
t
h
.
()
tc
s
c
hK
h
B
(2)
In Equation (2),
1
ce
K
K
，
ce
BB
.
The vecto
r
of
t
h
is
in the form of:
11
1
21
22
2
31
32
33
3
00
0
ts
ts
ts
xx
yy
zz
hh
kb
hk
k
h
b
kk
k
b
hh
(3)
At a certain
moment for a
fixed positio
n, and
thin
k that the magn
etic field stre
ngth and
the directio
n
is
con
s
tant.
Rotation
of t
he
sen
s
o
r
in
three
-
dim
e
n
s
ion
a
l
spa
c
e,
the id
eal
ou
tput
data within th
e spa
c
e of tra
j
ectory is
sph
e
rical.
2
2
t
hH
(4)
H
mean
s the lo
cation
of the magneti
c
fiel
d inten
s
it
y. Combinin
g Eq
uation (2)
with Equation
(4
),
we can get:
2
2
TT
T
ss
s
hA
h
b
A
h
b
A
b
H
(5)
c
T
c
K
K
A
c
bB
. Based
on t
he a
s
sumptio
n
of the elli
psoid comp
ensation ap
pro
a
ch co
nsi
d
e
r
s
the measure
m
ents of the actual out
p
u
t trajecto
ry to ellipsoid, na
mely Equation (5) said elli
psoi
d
equatio
n of
vector. So, t
he p
r
obl
em
of error
co
m
pen
sation
of
se
nsors int
o
ellip
soid
fitting
probl
em.
Cha
nging Eq
uation (5
) into
the general e
quati
on of qu
adri
c
su
rfa
c
e,
then we can
get:
22
12
3
4
5
2
67
8
9
1
0
(,
)
0
T
F
X
a
x
a
xy
a
y
a
x
z
a
yz
az
a
x
a
y
a
z
a
X
(6)
T
z
y
x
z
yz
xz
y
xy
x
X
]
1
[
2
2
2
T
a
a
a
a
]
[
10
3
2
1
，
The measurement data of ellipsoid fitting is to solv
e the coeffici
ent of ellipsoid. It is to meet
all
the sum
of th
e squa
re
s
of
the alg
e
b
r
aic dista
n
ce m
e
asu
r
em
ent d
a
ta to th
e elli
psoi
d mi
nimu
m
[9],a s
arg
m
i
n
(
)
E
.
2
1
2
)
,
(
D
X
F
E
N
i
i
(7)
T
N
X
X
X
X
D
3
2
1
,which is
a
6
N
matrix.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 4, April 2014: 3078 – 3
084
3080
In ord
e
r to
gu
arante
e
the
q
uadri
c
su
rface to ellip
soi
d
, to sati
sfy the
followin
g
con
s
traint
s
[10]:
0
4
0
)
det(
2
2
3
1
a
a
a
W
(8)
0
)
det(
)
(
3
1
A
a
a
(
9
)
3
2
2
1
2
/
2
/
a
a
a
a
W
，
6
5
4
5
3
2
4
2
1
2
/
2
/
2
/
2
/
2
/
2
/
a
a
a
a
a
a
a
a
a
A
，
As the f
r
ee p
a
ram
e
te
rs,
ca
n b
e
ch
osen
suitabl
e mag
n
ification, ma
king Equ
a
tion
(8) satisfy
1
4
2
2
3
1
a
a
a
.
1
T
C
(10)
Solution sati
sfy
the con
s
t
r
aint con
d
itio
ns (9
-10
)
of
the matrix E
quation
(7
),
usin
g L
agran
ge
multiplier met
hod availabl
e
:
13
1
()
d
e
t
(
)
0
T
T
DD
C
C
aa
A
(11)
Solving the Equation (11),
coeffici
ent
of ellipsoid for the least posi
t
i
ve characteri
stics of th
e
corre
s
p
ondin
g
eigenve
c
tors.
3. Impro
v
ed
Ellipsoid Fitting Method
Acco
rdi
ng to
the spe
c
ial
st
ructu
r
e of the
ma
trix, throu
gh the matrix
deco
m
po
siti
on, can
overcome the
defects of co
nstrai
nt matri
x
is singul
ar, and sim
p
lify
the feature ve
ctor to solve.
Firstly ,
]
[
2
1
D
D
D
,and
3
2
2
2
2
2
1
1
1
2
1
1
N
N
N
N
N
i
i
i
i
y
y
x
x
y
y
x
x
y
y
x
x
D
7
2
2
1
1
1
2
1
1
1
1
1
2
1
1
1
N
N
N
N
N
N
N
N
N
i
i
i
i
i
i
i
i
z
y
x
z
z
y
z
x
z
y
x
z
z
y
z
x
z
y
x
z
z
y
z
x
D
,
then
4
3
2
1
S
S
S
S
D
D
S
T
, and
1
1
1
D
D
S
T
，
2
1
2
D
D
S
T
，
T
T
S
D
D
S
2
1
2
3
，
2
2
4
D
D
S
T
.
We
ca
n get
t
he con
s
train matrix
4
3
2
1
C
C
C
C
C
,and
0
0
2
0
1
0
2
0
0
1
C
，
7
3
2
0
C
，
3
7
3
0
C
，
7
7
4
0
C
. Making
2
1
, and
T
a
a
a
3
2
1
1
，
10
9
8
7
6
5
4
2
a
a
a
a
a
a
a
.
Take th
e abo
ve matrix decompo
sition in
to (11), we ca
n get:
1
1
2
2
1
1
C
S
S
(12)
0
2
4
1
2
S
S
T
(13)
Whe
n
the sa
mpling data i
s
not in the same plan
e,
4
S
is a singul
ar m
a
trix [12], finishin
g availabl
e:
1
1
2
1
4
2
1
1
1
)
(
T
S
S
S
S
C
(14)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
The Approa
ch of Error
Cal
i
bration fo
r Three
-
a
x
i
s
Ma
gnetic
Hea
d
in
g Senso
r
(Xia
o Miaoxi
n
)
3081
1
2
1
4
2
T
S
S
(15)
Then Equ
a
tio
n
(10
)
ca
n be
chan
ged into
:
1
1
1
1
C
T
(16)
Matrix de
co
mpositio
n ab
ove co
mbine
Equation
(1
1)
with Equa
tion (1
4-1
6
)
solutio
n
, get
the
minimum co
rrespon
ding
e
i
genve
c
tors are
characte
ri
stic
ro
ot, an
d plu
g
in
Eq
uation
(9
). T
h
is
formula
(1
1)
solving th
e 1
0
d featu
r
e
vector i
n
to fo
rmula
(1
4)
solution of th
ree dim
e
n
s
io
nal
feature
vecto
r
to
de
crease
the
amou
nt
of cal
c
ul
ation
for
abo
ut a
third
of the
o
r
i
g
inal, a
nd
at
the
same time u
s
ing the impro
v
ed algorith
m
on accuracy
is co
nsi
s
tent
with the origi
nal algo
rithm.
Acco
rdi
ng to
the Equatio
n (5
-6) m
a
tri
x
A
、
b
can be
obtaine
d, an
d be
cau
s
e of
the
Eq.(10) for
amplificatio
n
co
efficient
, the matrix
is relative.
As the a
b
solut
e
value
of
A
magnification
by Equation (5), (6) corre
s
pondi
ng rel
a
tion ca
n be obt
ained:
)
/(
2
10
H
Ab
b
a
k
T
(17)
Cal
c
ulate
d
a
c
cordi
ng to the Equatio
n (17), matrix
A
、
b
，
, and by Equation (5) to
work
out the
corre
s
pondi
ng
relati
onship
betwe
en e
r
ror com
pen
sation
co
efficient m
a
tri
x
c
K
and
c
B
,
a
nd
compl
e
te the error comp
en
sation [11
-
14]
.
4. Soft
w
a
re
Simulation
In order to validate the above algo
rit
h
m,
simulati
on softwa
r
e.
Assumin
g
that the
magneti
c
se
n
s
or lo
catio
n
uniform mag
n
e
tic field,
the magneti
c
field stren
g
th of 0.52 Gau
ss.
We
divide the ideal output
t
h
spheri
c
al a
r
ea
into
N
regio
n
s, ea
ch seg
m
ented regio
n
, rando
m
sele
ction of a
measu
r
in
g p
o
int data. As sho
w
n in Fig
u
re 1
Figure 1. Plot of Reco
rde
d
Data
Set the s
e
ns
or error
c
oeffic
i
ent matrix is
res
p
ec
tively:
9681
.
0
3171
.
0
0392
.
0
0
0537
.
1
2221
.
0
0
0
1338
.
1
e
K
，
0016
.
0
0043
.
0
0159
.
0
e
B
.
Join the vari
a
n
ce i
s
0.000
3
gaussia
n
whi
t
e noise. At each lo
catio
n
reco
rd ide
a
l vector
t
h
and
error of sensor output
cont
ains ma
gneti
c
vecto
r
,
s
h
as shown in Figu
re 2.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 4, April 2014: 3078 – 3
084
3082
Figure 2. Tru
e
Magneti
c
Fi
eld Vecto
r
t
h
a
nd Erro
neo
us Measu
r
e
d
Vector
s
h
Usi
ng the
a
bove alg
o
rith
m to ellip
soi
d
, fitting and cal
c
ul
ation
error
com
p
ensation
c
oeffic
i
ent matrix
c
K
、
c
B
.
To test algo
rithm cal
c
ulati
on accu
ra
cy, with a give
n magn
etic
sen
s
o
r
attitude as a
ben
chma
rk, and the
com
puted e
r
ror
compen
satio
n
of the mag
n
e
t
ic se
nsor
co
urse. Re
co
rd
of
locatio
n
re
sp
ectively set t
he pitching A
ngle a
nd t
ilt
Angle, pitchin
g
Angl
e
and t
ilt Angle, pitching
Angle an
d tilt Angle, pitchi
n
g
Angle a
nd t
ilt A
ngle for t
he ro
und, e
a
c
h g
r
ou
p of u
n
iform record
3
6
points. Calcul
ate the yaw Angle
erro
r is shown in Figu
re 3:
Figure 3
.
Hea
d
ing Error of
Simulation
5. Experiment and Resul
t
Analy
s
is
To ide
n
tify the above
meth
od is
accu
rat
e
and
reli
able
,
and the
erro
r compe
n
sation test,
magneti
c
sen
s
or
wh
en in
st
allation of X and X axis of
turntable
accurate alig
nm
ent, Y in the X
axis turntabl
e
,
Y axis in the plane, the turnt
able from m
agneti
c
materials at the sa
me time.
First of all, in accord
an
ce
with the se
ct
i
on method, reco
rd the reg
i
onal samplin
g point
locatio
n
data.
Seco
ndly elli
psoi
d
coeffici
ent is calcula
t
ed u
s
ing
the
fitting metho
d
, as sho
w
n i
n
Table 1, and t
r
an
slated into
coefficie
n
t matrix and the error comp
en
sation
c
K
,
c
B
.
Table 1. Parameters of Ellipsoi
d
1
a
2
a
3
a
4
a
5
a
1.6060
-0.2900
1.5813
-0.0171
0.0130
6
a
7
a
8
a
9
a
10
a
1.7931
0.0142
0.0129
0.0191
-0.2704
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TELKOM
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ISSN:
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046
The Approa
ch of Error
Cal
i
bration fo
r Three
-
a
x
i
s
Ma
gnetic
Hea
d
in
g Senso
r
(Xia
o Miaoxi
n
)
3083
To test com
pen
sation eff
e
ct, re
spe
c
tively in the pitchin
g
Angle
0
and tilt Angle
0
,
pitchin
g
Angl
e
0
and
tilt Ang
l
e
30
, pitchin
g
A
ngle
30
and tilt
Angle
0
, pitchin
g
Angle
30
and
tilt Angle
30
for the state, the
use
of non
-m
agneti
c
turn
ta
ble a
s
a reference be
nch
m
ark,
within
the scop
e of the
0
-
360
of c
o
urse, interval record
10
headi
ng
sensor o
u
tput
data, the he
a
d
ing
error of me
a
s
uri
ng ma
gn
etic sensor.
Before an
d a
fter the com
pen
sation ya
w Angle e
r
ro
r is
sho
w
n in
Fig
u
re 4.
Comp
ensation bef
ore you
ca
n
see, the m
a
ximum erro
r o
f
the yaw An
gle
92
.
5
, compen
sati
on doe
s not e
x
ceed the ma
ximum error
of
0.
42
.
Figure 4, Erro
s at Differe
nt Points
6. Conclusio
n
Puts forward
the improved ellipsoid
based on th
e least sq
ua
re fitting method of
magneti
c
he
a
d
ing
sen
s
o
r
error
com
pen
sation m
e
tho
d
, throug
h an
alyzing th
e si
ngula
r
ity of the
con
s
trai
nt ma
trix, to overcome the in
st
ability pr
obl
e
m
of tradition
al ellipsoid fitting method,
and
the softwa
r
e
simulatio
n
an
d experime
n
tal validation. This meth
od can
comp
en
sate the sen
s
o
r
's
zero error, error
sen
s
itivity,
ortho
gonal
error
such
a
s
inhe
rent
error,
without
e
x
ternal
refere
nce
ben
chma
rk.
On the
non
-magneti
c
tu
rntable exp
e
ri
ments
sh
owe
d
that the
m
a
ximum e
rro
r o
f
magneti
c
he
a
d
ing is
not m
o
re tha
n
0.
42
, feasibility and a
c
cura
cy of thi
s
metho
d
is
verified by
the experim
e
n
ts.
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ces
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y o
n
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u
tomatic Ma
gn
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e
viati
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atio
n of M
a
g
neti
c
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adi
n
g
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n
a
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oma
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ng, Ho
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e
i
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j
uan
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l Ca
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c
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02-4
046
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NI
KA
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h
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l G
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din
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m
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