Accelerometer compensation in an inertial navigation system

Details Accelerometer compensation in an inertial navigation system Accelerometer compensation in an inertial navigation system

2000-10-10

2003-10-21

Noland, Thomas P. (Department: 2856)

Measuring and testing

Instrument proving or calibrating

Angle, direction, or inclination

C073S001380, C702S093000, C702S196000

Reexamination Certificate

active

06634207

ABSTRACT:

CROSS-REFERENCE TO RELATED APPLICATIONS
(Not applicable)
BACKGROUND OF THE INVENTION
This invention relates generally to inertial navigation systems and more specifically to compensating acceleration measurements for body frame rotation and establishing the optimal center of navigation for a general accelerometer configuration.
Rotation creates centripetal and tangential accelerations which are proportional to the lever arm from the center of rotation to the rotating body. Any rotation through an arbitrary origin should produce zero acceleration at the origin. In a practical inertial navigation system, the three accelerometers cannot share the same origin. As a result, rotation about a reference origin can produce erroneous sensed accelerations and compensation is required.
In the past, the lever arms associated with the tangential acceleration (angular acceleration dependent) terms were small and produced negligible errors compared to the performance requirements. Compensation was based on centripetal (angular rate) terms only.
Some inertial navigation systems use silicon accelerometers mounted on the block in positions where angular acceleration terms produce errors that can approach the performance requirements. In the present invention a complete size effect compensation technique is used that treats both angular rate and angular acceleration terms. The algorithm performance is determined analytically, and used to define the optimal center of navigation for a general accelerometer configuration.
The notation used herein is as follows: a vector is denoted with an overhead arrow, e.g. {right arrow over (w)}, while a matrix is denoted in boldface, e.g. C.
The direction cosine matrix can be represented analytically by
C
B
N
=exp(&egr; sin(
wt
)[{right arrow over (1)}
&phgr;
x
])  (1)
and the rate of change of the direction cosine matrix by
{dot over (C)}
B
N
=C
B
N
[{right arrow over (w)}x]
  (2)
which means the sinusoidal angular velocity is given by
{right arrow over (w)}=W
{right arrow over (1)}
&phgr;
  (3)
W=&egr;w
cos(
wt
)  (4)
and angular acceleration is given by
w
.
→
=
ⅆ
 
ⅆ
t
⁢
 
⁢
w
→
=
-
ε
⁢
 
⁢
w
2
⁢
 
⁢
sin
⁢
 
⁢
(
wt
)
⁢
 
⁢
1
→
φ
(
5
)
where &egr; is the magnitude of the angular displacement and w is the frequency about an arbitrary but fixed axis
{right arrow over (1)}
&phgr;
=[c
x
c
y
c
z
]  (6)
where
{square root over (
c
x
2
+c
y
2
+c
z
2
)}=1,  (7)
Accelerometers with lever arm vector {right arrow over (R)} will sense an acceleration
{right arrow over (a)}
B
=([
{right arrow over (w)}x
]
2
+[{dot over ({right arrow over (w)})}x
])
R
B
  (8)
where the cross product
{right arrow over (&phgr;)}x{right arrow over (R)}=[{right arrow over (&phgr;)}x]{right arrow over (R)}
  (9)
where
[
φ
→
⁢
 
⁢
x
]
=
[
0
-
φ
x
φ
y
φ
x
0
-
φ
x
-
φ
y
φ
x
0
]
(
10
)
and the cross product
{right arrow over (&phgr;)}x[{right arrow over (&phgr;)}x{right arrow over (R)}]=[{right arrow over (&phgr;)}x]
2
{right arrow over (R)}
  (11)
where
[
φ
→
⁢
 
⁢
x
]
2
=
[
-
φ
y
2
-
φ
x
2
φ
x
⁢
 
⁢
φ
y
φ
x
⁢
 
⁢
φ
z
φ
x
⁢
 
⁢
φ
y
-
φ
x
2
-
φ
z
2
φ
y
⁢
 
⁢
φ
z
φ
x
⁢
 
⁢
φ
z
φ
y
⁢
 
⁢
φ
z
-
φ
x
2
-
φ
y
2
.
]
(
12
)
If all three accelerometers shared the same origin, the lever arm R
B
=0, and for a navigation center defined as that origin, any angular rate through that origin would not produce measured acceleration in any axis.
The projections on to the X, Y, and Z axis are the body frame accelerations, i.e.
{right arrow over (a)}
x
=P
x
([
{right arrow over (w)}x]
2
+[{dot over ({right arrow over (w)})}x
])
{right arrow over (R)}
x
  (13)
{right arrow over (a)}
y
=P
y
([
{right arrow over (w)}x]
2
+[{dot over ({right arrow over (w)})}x
])
{right arrow over (R)}
y
  (14)
{right arrow over (a)}
z
=P
z
([
{right arrow over (w)}x]
2
+[{dot over ({right arrow over (w)})}x
])
{right arrow over (R)}
z
  (15)
where the projection matrices are
P
x
=
[
1
0
0
0
0
0
0
0
0
]
(
16
)
P
y
=
[
0
0
0
0
1
0
0
0
0
]
(
17
)
P
z
=
[
0
0
0
0
0
0
0
0
1
]
(
18
)
and {right arrow over (R)}
x
, {right arrow over (R)}
y
, and {right arrow over (R)}
z
are the lever arm vectors for accelerometers in the X, Y, and Z directions respectively.
This is a convenient way to write the measured acceleration, since the other elements of the vector are zero, i.e.
a
→
x
=
[
a
B
x
0
0
]
,
a
→
y
=
[
0
a
B
y
0
]
,
a
→
x
=
[
0
0
a
B
z
]
(
19
)
which can also be written as dot products, i.e.
a
Bx
={right arrow over (a)}
x
·{right arrow over (1)}
x
,a
By
={right arrow over (a)}
y
·{right arrow over (1)}
y
,a
Bz
={right arrow over (a)}
z
·{right arrow over (1)}
z
  (20)
where the unit vectors are defined as
1
→
x
=
[
1
0
0
]
,
1
→
y
=
[
0
1
0
]
T
,
1
→
x
=
[
0
0
1
]
T
(
21
)
Using the same matrix representations for cross products shown in Equation (10) and Equation (16) but substituting
{right arrow over (w)}=[w
x
w
y
w
z
]  (22)
{dot over ({right arrow over (w)})}=[{dot over (w)}
x
{dot over (w)}
y
{dot over (w)}
z
]  (23)
where
w
x
=Wc
x
  (24)
w
y
=Wc
y
  (25)
w
z
=Wc
z
  (26)
and for lever arms
{right arrow over (R)}
x
=[R
xx
R
yx
R
zx
]  (27)
{right arrow over (R)}
y
=[R
xy
R
yy
R
zy
]  (28)
{right arrow over (R)}
z
=[R
xz
R
yz
R
zz
]  (29)
where R
ij
is the i'th coordinate of the j'th accelerometer, the acceleration is given by
a
Bx
=(−
w
y
2
−w
z
2
)
R
xx
+(−
{dot over (w)}
z
+w
x
w
y
)
R
yz
+(
{dot over (w)}
y
+w
x
w
z
)
R
zx
  (30)
 
a
By
=(
{dot over (w)}
z
+w
x
w
y
)
R
xy
+(−
w
x
2
−w
x
2
)
R
yy
+(−
{dot over (w)}
x
+w
y
w
z
)
R
zy
  (31)
a
Bz
=(−
{dot over (w)}
y
+w
x
w
z
)
R
xz
+(
{dot over (w)}
x
+w
y
w
z
)
R
yz
+(−
w
x
2
−w
y
2
)
R
zz
  (32)
where
{right arrow over (a)}
B
=[a
Bx
a
By
a
Bz
]tm (33)
The lever arms are a measurement of the distances between the accelerometers and the center of navigation.
The direction cosine matrix is used to convert the body (instrument) frame data to the navigation frame, i.e.
{right arrow over (a)}
N
=C
B
N
{right arrow over (a)}
B
.  (34)
The acceleration in the body frame (see Equations (30)-(32)) is a function of [{right arrow over (w)}x]
2
which is second order in &egr; and [{dot over ({right arrow over (w)})}x] which is first order in &egr;. Therefore, to convert to the navigation frame, only a first order expansion of the direction cosine matrix (see Equation (1)) is required, i.e.
C
B
N
≈I
+&egr; sin(
wt
)[{right arrow over (1)}
&phgr;
x
].  (35)
The full matrix multiplication in Equation (34) is not required as only DC terms are of interest. The mean acceleration in the navigation frame is given by
 
<a
Nx
>=½&egr;
2
w
2
[c
y
2
(
R
xz
−
R
xx
)+
c
x
2
(
R
xy
−
R
xx
)+
c
x
c
y
(
R
yx
−R
yz
)+
c
x
c
z
(
R
zx
−R
zy
)]  (36)
<a
Ny
>=½&egr;
2
w
2
[c
x
2
(
R
yz
−
R
yy
)+
c
z
2
(
R
yz
−
R
yy
)+
c
x
c
y
(
R
xy
−R
xz
)+
c
y
c

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