Aerospace and Electronic Systems Magazine August 2017 - 43
Nusbaum and Klein
full model of A (without our assumptions) was analyzed in [2].
There, the well-known Schuler modes and two real eigenvalues,
which are the diverging and converging modes associated with
the INS vertical channel, are found. The eigenvalues of D are decoupled from those of A and are determined by the geometric arrangement of the accelerometers and the residuals model. Recall
that from (13), it seems that the performance of the system is influenced by the dynamics of the accelerometers through the D block.
Looking at (13), we derive the structure as
δω n hω 2
=
b 0
hω1 δωibb
0 b
(21)
GF FA OBSERVABILITY ANALYSIS
The FA process is used to give a better estimation of Tbn than the
one obtained by the coarse alignment phase and in the process
to estimate the IMU residuals. FA usually involves a state space
estimation algorithm, such as the extended Kalman filter (EKF),
with external measurements (usually external velocity updates).
Because the alignment occurs at a known location, position-state
detectability is not an issue; therefore, the position error states can
be removed from (13). Thus, after eliminating the position error
states, (13) reduces to
which is a linear system (decoupled). Because the state that relates
to the biases has no dynamics, we have N zero eigenvalues, and
the rest of the eigenvalues are determined by the first row of (21),
which is the following LTI system (for a stationary INS):
δ v n 0 − f n × 0 hy1 δ v n hy1 0
n
n
I 0 δ + 0 0 wy
0
δ = 0
b
b
δω
0
0 hω1 δωib hω1 0 wb
0
b 0
0
0 0 b 0 I
δω = hω 2δω + hω1b
where we assume for simplicity of the derivation that Tbn = I.
n
(22)
b
ib
For this system, the biases are now the inputs and the dynamics
are driven through the properties of hω2. This structure suggests
that in absence of a bias, if the eigenvalues of hω2 are located inside
the left half-plane the GF-INS will self-correct any initial error and
will produce (in steady state) zero error in the angular velocity error states. At best, we would expect that the use of GF-INS will
have no negative effect on the overall system, which means that
because we measure accelerations, only angular acceleration can
be estimated directly, can be seen in (3); thus, integration is required for the angular velocity estimation, which in turn is linearly
related to time. Therefore, the eigenvalues of hω2 will be located at
the origin at best. If not, i.e., if the eigenvalues are inside the right
half-plane (RHP), then it is possible to search for the accelerometers' best arrangement in the sense of angular velocity estimation
error propagation.
Solving of (22) yields the eigenvalues of the angular velocity
error states. Herein, we examine two GF configurations for solving
the eigenvalues problem, where the stationary GF-INS is located
on the equator. Plugging (16) and (17) into the configuration matrix H, calculating h from (3), and using the matrix hω2 as the upper
three rows of hA yield the eigenvalues through (22):
λ1,2 = 0, λ3 = 5.3144 ·10−9 ≅ ( 7.29 ·10−5 ) = ωie2
2
(23)
Repeating the procedure with the 12-accelerometer configuration
using (18) and (19) gives
λ1 = 2.126 ·10−9 , λ2 = 3.189 ·10−9 , λ3 = ωie2
(24)
For both cases examined, the eigenvalues of hω2 were located,
as expected, in the RHP. From a control engineering viewpoint,
this suggest that performance can be enhanced through better positioning of the accelerometers. In addition, the eigenvalues were
independent of the radius of the circle for the nine-accelerometer
configuration and of the cube length for the 12-accelerometer configuration.
AUGUST 2017
(25)
F1
OBSERVABILITY ANALYSIS
We employ velocity measurements for the FA process in which the
measurement model is given by the following:
δ vn
n
δ
δ z = I 03×( 6 + N ) b
δωib
C
b
(26)
Again, we have a block diagonal dynamic model, which suggests
that the dynamics of the accelerometers are not affected by the
dynamics of the entire system. To examine the observability of the
systems in (25) and (26), we construct the observability matrix [9].
The resultant observability matrix is as follows:
0
0
0
I
C
n
0 − f ×
0
h y1
CF
1
2
n
CF1 0
− f ×
0
0
OA
=
=
=
3
CF
1 0
− f n × hω1 OB
0
0
n −1 0
0
0
0
CF1
(27)
This special structure of the observability matrix arises because
F14 = 0; therefore, all rows from the fifth row to the last one are
nullified. Thus, the observability matrix rank is determined only
by the rank of OA.
Recall that while stationary, rank f × = 2. This suggests that
in the third block row, one row is zero, and in the last block row,
another row is zero. In addition, hy1 and hω1 must be full rank; this requirement arises from the geometric arrangement requirements of the
accelerometer cluster. This leads to an understanding that the observability matrix rank is always 10, regardless of the GF configuration.
IEEE A&E SYSTEMS MAGAZINE
(
)
43
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