Aerospace and Electronic Systems Magazine August 2017 - 44
Control Theoretic Approach to Gyro-Free Inertial Navigation Systems
Nine-Accelerometer Configuration
OBSERVABLE AND UNOBSERVABLE SUBSPACES
The observable and unobservable subspaces are spanned by Im ( )
and ker ( ) respectively, where the state space can be spanned using Im ( ) ⊕ ker ( ). A state transformation matrix T is to be defined to derive the observable and unobservable subspaces. To that
end, we take 10 independent rows from the observability matrix
(its rank). Those rows represent Im ( ) and are noted as 1. To
complete T, five independent vectors orthogonal to 1 are required.
Those vectors are null (1 ). While 1 is constructed using all rows
from , which are not zero, ker ( ) is constructed using null ( O ).
The state transformation matrix is then as follows:
Following the same procedure we used previously, we get
Cube-Shaped 12-Accelerometer Configuration
vn
ve
vd
1
b1 + b4 + b7 − e g
3
1
b2 + b5 + b8 + n g
3
1
y9 =
b3 + b6 + b9
3
−ω2 g
ω1 g
g
b6 − b9
3d
g 2b − b − b
3d 3 6 9
For the 12-accelerometer GF-INS, we obtain where for clarity, the
left column contains observable states and linear combinations of
them and the right column contains unobservable states. Observing (30), we see that the velocity error states are observable, as
expected from the velocity measurements. In addition, ϵd and δω3
are not observable as in classical INS, where ϵd and the z-axis gyro
bias are not observable.
where the 10 observable states and linear combination of them are
located in the left column and the unobservable states are located
in the right column. Observing (31), we see that the velocity error
states are observable; in addition, ϵd and δω3 aren't observable as in
the 12-accelerometer case.
T = 1
2
(28)
Let yO be the original error-state vector as given in (13). The new
state vector representing the observable and unobservable states is
as follows:
yn = Tyo
(29)
RESULTS
δ vn
δ ve
δ vd
1 7b − b − b + 7b + 3b + 3b + 3b − b − b + 3b − b − b − g
4
5
6
7
8
9
10
11
12
e
20 1 2 3
1 7b − b − b − b + 3b − b + 3b + 7b + 3b − b + 3b − b + g
5
6
7
8
9
10
11
12
n
20 2 1 3 4
1 7b − b − b − b − b + 3b − b − b + 3b + 3b + 3b + 7b
6
7
8
9
10
11
12
3 2 1 4 5
y12 = 20
−δω2 g
δω1 g
g
1
5
1
1
5
1
b1 − b3 + b4 + b5 + b6 + b7 − b9 − b10 − b11 − b12
2
2
2
2
2
2
5d
g b − b + 1 b − 1 b + 1 b + b + 5 b − 1 b − 5 b − b
5d 2 3 2 5 2 6 2 7 8 2 9 2 10 2 11 12
44
ω3
1
b1 +
e
3g
1
b2 −
n
3g
1
b4 +
e
3g
1
b5 −
n
3g
1
e
b7 +
3g
1
n
b8 −
3g
d
b4 − b1
1
b5 − b2 + b1
2
1
1
b6 − b2 + 7b2 − b3 − e
2
g
1
b7 − b2 + b1
2
b8 − b2
1
1
b9 − 7b2 + b1 − b3 + n
2
g
1
1
b10 + 5b1 + b2 + e
2
g
1
1
b11 + b1 + 5b2 − n
2
g
b12 − b3
(31)
d
δω3
IEEE A&E SYSTEMS MAGAZINE
(30)
AUGUST 2017
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