Apparatuses and methods for calibrating magnetometer attitude-independent parameters
View Patent ↗Methods and apparatuses for calibrating attitude-independent parameters of a 3-D magnetometer are provided. A calibration method includes storing and updating data related to a N×9 matrix T and a N×1 matrix U extended for each measurement with an additional row and an additional element, respectively, the additional row and the additional element being calculated based on values measured by the 3-D magnetometer for the respective measurement. The method further includes calculating analytically (1) a symmetric non-orthogonal 3×3 matrix D representing scaling and skew of the 3-D magnetometer measured values and (2) a vector b representing bias of the 3-D magnetometer measured values, using the stored data and a singular value decomposition (SVD) method.
1. A method for determining an orientation of a 3-D device, by calibrating attitude -independent parameters of a 3-D magnetometer attached to the 3-D device, the method comprising:
receiving movement data of the 3-D device via the 3-D magnetometer;
storing and updating data related to a T matrix of size N×9 and a U matrix of size N×1, each of the N rows of matrix T corresponding to a respective measured value of the 3-D magnetometer and comprising values calculated based on a first set of 9 functions of the respective measured value by the 3-D magnetometer, wherein N is a number of sample measurements by the 3-D magnetometer, and each of the N rows of matrix U corresponding to a respective measured of the 3-D magnetometer and comprising a value calculated based on a second function of the respective measured value by the 3-D magnetometer;
calculating analytically (1 )a symmetric nonorthogonal 3×3 matrix D representing scaling and skew of each axis of the 3-D magnetometer measured values and (2) a vector b representing bias of each axis of the 3-D magnetometer measured values, using the stored data and a singular value decomposition method; and wherein the calculating comprises:
computing a vector K=(T T ×T) −1 ×T T ×U;
generating a matrix F using elements of the vector K, and determining a matrix G, where F=G×G, using a singular value decomposition method;
determining a scale offset bias
{tilde over (b)}= G −1 ×[ K (6) K (7) k (8)] T
determining a scale factor
co
=
H
2
[
K
(
6
)
K
(
7
)
K
(
8
)
]
×
F
-
1
×
[
K
(
6
)
K
(
7
)
K
(
8
)
]
T
-
K
(
9
)
where |H| 2 is a square of local geomagnetic field strength;
calculating the vector b =sqrt(co) {tilde over (b)}and matrix pD =sqrt(co)·G ; and
extracting the symmetric non-orthogonal 3×3 matrix D=pD−1 3×3 , wherein K(i) represents the i th element of vector K
further wherein a 9×9 matrix L is used to accumulate T T ×T , and a 9×1 matrix M is used to accumulate T T ×U, wherein at time step n+1 , matrix L and matrix M are updated according to:
L n+1 = L n +( T n+1 T × T n+1 )
M n+1 = M n +( T n+1 T × U n+1 )
wherein T n+1 ,is an element at n+1 row of T, and U n+1 , is an element at n+1 row of U, and are functions of a magnetometer sample measurement at a current time step;
calibrating attitude-independent parameters of a 3-D magnetometer attached to the 3-D device using matrix D to compensate for soft iron effect and matrix b to compensate for hard iron effect; and
determining an orientation of the device using the attitude-independent parameters.
2. The method of claim 1 , wherein
F
=
[
1
+
K
(
1
)
+
K
(
2
)
K
(
3
)
K
(
4
)
K
(
3
)
1
+
K
(
1
)
-
2
K
(
2
)
K
(
5
)
K
(
4
)
K
(
5
)
1
-
2
K
(
1
)
+
K
(
2
)
]
.
3. The method of claim 1 , wherein
T
=
[
[
B
x
2
+
B
y
2
-
2
B
z
2
B
x
2
-
2
B
y
2
+
B
z
2
2
B
x
·
B
y
2
B
x
·
B
z
2
·
B
y
·
B
z
-
2
B
x
-
2
B
y
-
2
B
z
1
]
1
…
[
B
x
2
+
B
y
2
-
2
B
z
2
B
x
2
-
2
B
y
2
+
B
z
2
2
B
x
·
B
y
2
B
x
·
B
z
2
·
B
y
·
B
z
-
2
B
x
-
2
B
y
-
2
B
z
1
]
N
]
and
U
=
[
-
(
B
x
2
+
B
y
2
+
B
z
2
)
1
…
-
(
B
x
2
+
B
y
2
+
B
z
2
)
N
]
,
wherein B x is a measurement of axis x, B y is a measurement of axis y, and B z is a measurement of axis z.
4. The method of claim 1 , further comprising estimating accuracy of determining the symmetric non-orthogonal 3×3 matrix D and the vector b.
5. The method of claim 4 , wherein the accuracy of determining the symmetric non-orthogonal 3×3 matrix D and the vector b is estimated by computing matrices
Y
1
=
[
0
0
0
2
3
pD
11
0
-
2
3
pD
33
2
3
pD
12
0
-
2
3
pD
23
0
0
0
2
3
pD
11
-
2
3
pD
22
0
0
2
3
pD
13
-
2
3
pD
23
0
0
0
pD
12
pD
12
0
pD
11
+
pD
22
pD
23
pD
13
0
0
0
pD
13
0
pD
13
pD
23
pD
11
+
pD
33
pD
12
0
0
0
0
pD
23
pD
23
pD
13
pD
12
pD
22
+
pD
33
pD
11
pD
12
pD
13
b
x
0
0
b
y
b
z
0
pD
12
pD
22
pD
23
0
b
y
0
b
x
0
b
z
pD
13
pD
23
pD
33
0
0
b
z
0
b
x
b
y
2
b
x
2
b
y
2
b
z
0
0
0
0
0
0
]
Y
2
=
K
×
[
0
0
0
4
3
pD
11
4
3
pD
22
4
3
pD
33
2
pD
12
2
pD
13
2
pD
23
]
∂
K
∂
J
=
1
co
·
(
Y
1
-
Y
2
)
A
=
A
+
(
T
n
+
1
T
×
T
n
+
1
)
and
P
KK
=
σ
z
2
·
A
-
1
;
calculating an error covariance matrix
P
JJ
=
(
∂
K
∂
J
)
-
1
×
P
KK
×
(
∂
K
∂
J
)
-
1
;
and
computing error of the elements of b and D as elements of J=[b x b y b z pD 11 pD 22 pD 33 pD 12 pD 13 pD 23 ] T according to ε J =sqrt(diag(P JJ ))
wherein
pD
=
I
3
×
3
+
D
=
[
pD
11
pD
12
pD
13
pD
12
pD
22
pD
23
pD
13
pD
23
pD
33
]
.
6. An apparatus for determining an orientation of a 3-D device, wherein the apparatus is configured to perform a calibration of attitude-independent parameters of a 3-D) magnetometer attached to the 3-D device, comprising:
an interface configured to receive field values measured on each of three axes by the 3-D magnetometer; and
a data processing unit configured
to receive movement data of the 3-D device via the 3-D magnetometer,
to store and update data related to a T matrix of size N×9 and a U matrix of size N×1 ,each of the N rows of matrix T responding to a respective measured value of the 3-D magnetometer and comprising values calculated based on a first set of 9 functions of the respective measured value by the 3-D magnetometer, each of the N rows of matrix U corresponds to a respective measured value by the 3-D magnetometer and comprises a value calculated based on a second function of the respective measured value by the 3-D magnetometer, wherein N is a number of sample measurements by the 3-D) magnetometer,
to calculate analytically (1) a symmetric non-orthogonal 3×3 matrix D representing scaling and skew of each axis of the 3-D magnetometer measured values and (2) a vector b representing bias of each axis of the 3-D magnetometer measured values, using the stored data and a singular value decomposition method, and
wherein the data processing unit is configured to calculate the symmetric non-orthogonal 3×3 matrix D and the vector b
by: (1) computing a vector K=(T T ×T) −3 ×T T ×U;
(2) generating a matrix F using elements of the vector K, and determining a matrix G, where F=G×G, using a singular value decomposing method;
(3) determining a scale offset bias
{tilde over (b)}= G −1 ×[ K (6) K (7) k (8)] T
(4) determining a scale factor
co
=
H
2
[
K
(
6
)
K
(
7
)
K
(
8
)
]
×
F
-
1
×
[
K
(
6
)
K
(
7
)
K
(
8
)
]
T
-
K
(
9
)
where |H| 2 a square of local geomagnetic field strength;
(5) calculating the vector b =sqrt(co) {tilde over (b)}and matrix pD =sqrt(co)·G ; and
(6) extracting the symmetric non-orthogonal 3×3 matrix D =pD−I 3×3 , wherein K(i) represents the i th element of vector K,
further wherein the data processing unit is configured to use a 9×9 matrix L is to accumulate T T ×T , and a 9×1 matrix M to accumulate T T ×U , wherein at time step n+1, matrix L and matrix M are updated according to:
L n+1 = L n +( T n+1 T × T n+1 )
M n+1 = M n +( T n+1 T × U n+1 )
wherein T n+1 , is an element at n+ 1 row of T, and U n+1 , is an element at n+1 row of U, and are functions of a magnetometer sample measurement at a current time step;
to calibrate attitude-independent parameters of a 3-D magnetometer attached to the 3-D) device using matrix D to compensate for soft iron effect and matrix b to compensate for hard iron effect; and
to determine an orientation of the device using the attitude-independent parameters.
7. The method of claim 6 further comprising:
determining if the 3-D magnetometer measured values meet a predetermined criterion; and
using the 3-D magnetometer measured values in the updating of the matrices T and U only if they meet the predetermined criterion.
8. The apparatus of claim 7 , where
F
=
[
1
+
K
(
1
)
+
K
(
2
)
K
(
3
)
K
(
4
)
K
(
3
)
1
+
K
(
1
)
-
2
K
(
2
)
K
(
5
)
K
(
4
)
K
(
5
)
1
-
2
K
(
1
)
+
K
(
2
)
]
.
9. The apparatus of claim 7 , wherein
T
=
[
[
B
x
2
+
B
y
2
-
2
B
z
2
B
x
2
-
2
B
y
2
+
B
z
2
2
B
x
·
B
y
2
B
x
·
B
z
2
·
B
y
·
B
z
-
2
B
x
-
2
B
y
-
2
B
z
1
]
1
…
[
B
x
2
+
B
y
2
-
2
B
z
2
B
x
2
-
2
B
y
2
+
B
z
2
2
B
x
·
B
y
2
B
x
·
B
z
2
·
B
y
·
B
z
-
2
B
x
-
2
B
y
-
2
B
z
1
]
N
]
and
U
=
[
-
(
B
x
2
+
B
y
2
+
B
z
2
)
1
…
-
(
B
x
2
+
B
y
2
+
B
z
2
)
N
]
wherein B x is a measurement of axis x, B y is a measurement of axis y, and B z is a measurement of axis z.
10. The apparatus of claim 7 , wherein the data processing unit is further configured to estimate accuracy of determining the symmetric non-orthogonal 3×3 matrix D and the vector b.
11. The apparatus of claim 10 , further wherein the data processing unit estimates the accuracy of determining the symmetric non-orthogonal 3×3 matrix D and the vector b is estimated by computing matrices
Y
1
=
[
0
0
0
2
3
pD
11
0
-
2
3
pD
33
2
3
pD
12
0
-
2
3
pD
23
0
0
0
2
3
pD
11
-
2
3
pD
22
0
0
2
3
pD
13
-
2
3
pD
23
0
0
0
pD
12
pD
12
0
pD
11
+
pD
22
pD
23
pD
13
0
0
0
pD
13
0
pD
13
pD
23
pD
11
+
pD
33
pD
12
0
0
0
0
pD
23
pD
23
pD
13
pD
12
pD
22
+
pD
33
pD
11
pD
12
pD
13
b
x
0
0
b
y
b
z
0
pD
12
pD
22
pD
23
0
b
y
0
b
x
0
b
z
pD
13
pD
23
pD
33
0
0
b
z
0
b
x
b
y
2
b
x
2
b
y
2
b
z
0
0
0
0
0
0
]
Y
2
=
K
×
[
0
0
0
4
3
pD
11
4
3
pD
22
4
3
pD
33
2
pD
12
2
pD
13
2
pD
23
]
∂
K
∂
J
=
1
co
·
(
Y
1
-
Y
2
)
A
=
A
+
(
T
n
+
1
T
×
T
n
+
1
)
and
P
KK
=
σ
z
2
·
A
-
1
;
calculating an error covariance matrix
P
JJ
=
(
∂
K
∂
J
)
-
1
×
P
KK
×
(
∂
K
∂
J
)
-
1
;
and
computing error of the elements of b and D as elements of J=[b x b y b z pD 11 pD 22 pD 33 pD 12 pD 13 pD 23 ] T according to ε J =sqrt(diag(P JJ ))
wherein
pD
=
I
3
×
3
+
D
=
[
pD
11
pD
12
pD
13
pD
12
pD
22
pD
23
pD
13
pD
23
pD
33
]
.
12. The apparatus of claim 7 wherein the data processing unit is further configured to:
determine if the 3-D magnetometer measured values meet a predetermined criterion; and
use the 3-D magnetometer measured values in the updating of the matrices T and U only if they meet the predetermined criterion.
13. A non-transitory computer readable medium storing executable codes, which, when executed by a processor, make the processor execute a method for determining an orientation of a 3-D device by calibrating attitude-independent parameters of a 3-D magnetometer attached to the 3-D device, the method comprising
receiving movement data of the 3-D device via the 3-D magnetometer;
storing and updating data related to a T matrix of size N×9 and a U matrix of size N×1, each of the N rows of matrix T corresponding to a respective measured value of the 3-D magnetometer and comprising values calculated based on a first set of 9 functions of the respective measured value by the 3-D magnetometer, wherein N is a number of samples measurements by the 3-D magnetometer, and each of the N rows of matrix U corresponding to a respective measured vale of the 3-D magnetometer and comprising a value calculated based on a second function of the respective measured value by the 3-D magnetometer;
calculating analytically (1) a symmetric non-orthogonal 3×3 matrix D representing scaling and skew of each axis of the 3-D magnetometer measured values and (2) a vector b representing bias of each axis of the 3-D magnetometer measured values, using the stored data and a singular value decomposition method;
wherein the calculating comprises:
computing a vector K=(T T ×T) −1 ×T T ×U;
generating a matrix F using elements of the vector K, and determining a matrix G, where F=G×G, using a singular value decomposition method:
determining a scale offset bias
{tilde over (b)}= G −1 ×[ K (6) K (7) k (8)] T
determining a scale factor
co
=
H
2
[
K
(
6
)
K
(
7
)
K
(
8
)
]
×
F
-
1
×
[
K
(
6
)
K
(
7
)
K
(
8
)
]
T
-
K
(
9
)
where |H| 2 is a square of local geomagnetic field strength;
calculating the vector b =sqrt(co)·{tilde over (b)}and matrix pD =sqrt(co)·G ; and
extracting the symmetric non-orthogonal 3×3 matrix D=pD−I 3×3 , wherein K(i) represents the i th element of vector K,
wherein a 9×9 matrix L is used to accumulate T T×T, and a 9×1 matrix M is used to accumulate T T ×U , wherein at time step n+ 1 , matrix L and matrix M are updated according to:
L n+1 = L n +( T n+1 T × T n−1 )
M n+1 = M n +( T n+1 T × U n−1 )
further wherein T n+1 is an element at n+1 row of T, and U n+1 , is an element at n+1 row of U, and are functions of a magnetometer sample measurement at a current time step;
calibrating attitude-independent parameters of a 3-D magnetometer attached to the 3-D device using matrix D to compensate for soft iron effect and matrix b to compensate for hard iron effect and
determining an orientation of the device using the attitude-independent parameters.
14. The computer readable medium of claim 13 , where
F
=
[
1
+
K
(
1
)
+
K
(
2
)
K
(
3
)
K
(
4
)
K
(
3
)
1
+
K
(
1
)
-
2
K
(
2
)
K
(
5
)
K
(
4
)
K
(
5
)
1
-
2
K
(
1
)
+
K
(
2
)
]
.
15. The computer readable medium of claim 13 , wherein
T
=
[
[
B
x
2
+
B
y
2
-
2
B
z
2
B
x
2
-
2
B
y
2
+
B
z
2
2
B
x
·
B
y
2
B
x
·
B
z
2
·
B
y
·
B
z
-
2
B
x
-
2
B
y
-
2
B
z
1
]
1
…
[
B
x
2
+
B
y
2
-
2
B
z
2
B
x
2
-
2
B
y
2
+
B
z
2
2
B
x
·
B
y
2
B
x
·
B
z
2
·
B
y
·
B
z
-
2
B
x
-
2
B
y
-
2
B
z
1
]
N
]
and
U
=
[
-
(
B
x
2
+
B
y
2
+
B
z
2
)
1
…
-
(
B
x
2
+
B
y
2
+
B
z
2
)
N
]
wherein B x is a measurement of axis x, B y is a measurement of axis y, and B z is a measurement of axis z.
16. The computer readable medium of claim 13 , wherein the method further comprises estimating accuracy of determining the symmetric non-orthogonal 3×3 matrix D and the vector b.
17. The computer readable medium of claim 16 , wherein the method further comprising wherein the accuracy of determining the symmetric non-orthogonal 3×3 matrix D and the vector b is estimated by computing matrices
Y
1
=
[
0
0
0
2
3
pD
11
0
-
2
3
pD
33
2
3
pD
12
0
-
2
3
pD
23
0
0
0
2
3
pD
11
-
2
3
pD
22
0
0
2
3
pD
13
-
2
3
pD
23
0
0
0
pD
12
pD
12
0
pD
11
+
pD
22
pD
23
pD
13
0
0
0
pD
13
0
pD
13
pD
23
pD
11
+
pD
33
pD
12
0
0
0
0
pD
23
pD
23
pD
13
pD
12
pD
22
+
pD
33
pD
11
pD
12
pD
13
b
x
0
0
b
y
b
z
0
pD
12
pD
22
pD
23
0
b
y
0
b
x
0
b
z
pD
13
pD
23
pD
33
0
0
b
z
0
b
x
b
y
2
b
x
2
b
y
2
b
z
0
0
0
0
0
0
]
Y
2
=
K
×
[
0
0
0
4
3
pD
11
4
3
pD
22
4
3
pD
33
2
pD
12
2
pD
13
2
pD
23
]
∂
K
∂
J
=
1
co
·
(
Y
1
-
Y
2
)
A
=
A
+
(
T
n
+
1
T
×
T
n
+
1
)
and
P
KK
=
σ
z
2
·
A
-
1
;
calculating an error covariance matrix
P
JJ
=
(
∂
K
∂
J
)
-
1
×
P
KK
×
(
∂
K
∂
J
)
-
1
;
and
computing error of the elements of b and D as elements of J=[b x b y b z pD 11 pD 22 pD 33 pD 12 pD 13 pD 23 ] T according to ε J =sqrt(diag(P JJ )),
wherein
pD
=
I
3
×
3
+
D
=
[
pD
11
pD
12
pD
13
pD
12
pD
22
pD
23
pD
13
pD
23
pD
33
]
.
18. The computer readable medium of claim 13 wherein the method further comprises:
determining if the 3-D magnetometer measured values meet a predetermined criterion; and
using the 3-D magnetometer measured values in the updating of the matrices T and U only if they meet the predetermined criterion.