Aerospace and Electronic Systems Magazine August 2017 - 42

Control Theoretic Approach to Gyro-Free Inertial Navigation Systems

Figure 1.

GF configurations: (A) 9 accelerometers and (B) 12 accelerometers.

GF CONFIGURATIONS

and the sensitivity axes are

Two GF configurations are addressed. Because accelerometers often come in triads, the configurations chosen are triad based. The
chosen configurations are as follows:
1. Nine accelerometers on a plane, evenly distributed on a circle
with radius d (the angle between each pair is 120°), as presented in Figure 1A. Accelerometer sensitivity axes are coaligned.
The sensitivity axes of the accelerometers are aligned with the
inertial frame axes. The positions are given by [27]

3
0 0 0
2


1
P = d 1 1 1 −
2

0 0 0 0



3
2
1
−
2
0

3
2
1
−
2
0

3
2
1
−
2
0

−

3
2
1
−
2
0

−

3

2 
1 
− 
2 
0 



−

(16)

and the orientations are given by
T

1 0 0 1 0 0 1 0 0 


E = 0 1 0 0 1 0 0 1 0  .
0 0 1 0 0 1 0 0 1 

(17)

2. Twelve accelerometers on and inside a cube, as shown in Figure 1B. One triad is located at the origin, and the others are
located on the side panes of the cube. Accelerometer sensitivity
axes are coaligned. For a cube with length 2d, the locations of
the accelerometers are given by [30]
0 0 0 1 1 1 0 0 0 0 0 0 


P = d 0 0 0 0 0 0 1 1 1 0 0 0 
0 0 0 0 0 0 0 0 0 1 1 1 

42

(18)

T

1 0 0 1 0 0 1 0 0 1 0 0 


E = 0 1 0 0 1 0 0 1 0 0 1 0  .
0 0 1 0 0 1 0 0 1 0 0 1 

(19)

EIGENVALUES AND STABILITY ANALYSIS
From control point of view, the first step in analyzing a dynamic
system model, as in the GF-INS error model (14), is to check its
stability. In this case, we have a linear time-varying system, because the A matrix (14) depends on the specific force that acts on
the body through  f b × and that force can change through time.
In addition, the matrices D and B are time varying because of dependency on angular velocity through the block matrices in (7).
However, for a stationarity INS and when neglecting the Earth turn
rate, these matrices are constant while hy2 = 0 and hω2 = 0, and the
system then becomes linear time invariant (LTI). The stability of
an LTI system can be determined by examining its eigenvalues.
Because A and D, defined in (14), are square matrices, we can
write the eigenvalues problem as follows:
 λ I − A
B

det ( λ I − F ) = 0 → det   9
  = 0

λ
−
I
D
0
3+ n


→ det ( λ I 9 − A ) det ( λ I 3 + n − D ) = 0

(20)

The eigenvalues of A are also the eigenvalues of F, and the
same goes for D. Observation of A reveals that it's a diagonal matrix; therefore, its eigenvalues lie on its diagonal. Because all elements on A's diagonal are zeros, we get nine zeros as eigenvalues,
which translates into integrators dynamics. The eigenvalues of the

IEEE A&E SYSTEMS MAGAZINE

AUGUST 2017



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