Aerospace and Electronic Systems Magazine August 2017 - 41
Nusbaum and Klein
is the matrix formed by taking the lower three rows of h, and hy2 is
the matrix formed by taking the lower three rows of hA.
GF KINEMATICS
Classical INS kinematic equations are usually expressed in the
navigation frame and consist of the following states [26]:
1. pLLH-position vector (latitude ϕ, longitude λ, height h).
2. vebn -velocity vector expressed in navigation frame (vn north, ve
east, and vd down).
3. Tbn-the transformation matrix between the body and the navigation frames (body to navigation).
p LLH
vn
ve
=
+
− vd
RM + h ( RN + h ) cos (φ )
(
T
)
where RM is the meridian local radius of curvature, RN is the normal local radius of curvature, Ωien is the skew symmetric form of
the Earth rate angular velocity vector expressed in the navigation
n
frame, and Ωen
is the skew symmetric form of the navigation frame
rate relative to the ECEF frame angular velocity vector expressed
in the navigation frame. The inputs to (8) are the classical IMU
readings: specific force from the accelerometers and angular velocities from the gyros ωibb . In a GF-INS, the specific force obtained
from (3) is plugged into the classical navigation equation of motion
(8), yet the required gyro output is not available. Instead, ωibb is
obtained by integration of ω ibb defined in (3). Therefore, the angular
velocity vector is added to the INS kinematic equations. Thus, the
GF-INS kinematic equations (which has three additional states)
applicable for any GF configuration are as follows:
(
T
)
n
venn = Tbn ha Y − M + g ibn − Ωen
+ 2Ωien vebn
(
T = T Ω − Ω + Ω
n
b
n
b
b
ib
n
ie
n
en
)
0
0 δ p n 0
0 −T f × T h δ v n Tbn hy1
+
δY
n
0
0
Tbn δ 0
b
0
0
hω 2 δω hω1
I
n
b
b
n
b y2
(11)
Let the accelerometer residuals be defined by a time-varying bias
b(t) and a Gaussian white noise wy:
δ Y = b ( t ) + wy , b ( t ) = wb
(12)
Augmenting (12) with the error-state model (11) entails the GFINS error-state model
n
Tbn = Tbn Ωbib − Ωien + Ωen
Tbn
vn
ve
+
− vd
p LLH =
R
h
R
h
cos
φ
+
+
( N ) ( )
M
δ p n 0
n
δ v = 0
δ n 0
b
δω 0
(8)
)
n
+ 2Ωien vebn
venn = Tbn f ibb + g ibn − Ωen
(
Augmenting the angular velocity error-state model (10) with
the INS error-state model (7) gives the following:
(9)
n
b
T
ω = hω Y − M n
b
ib
δ p n 0
n
δ v 0
δ n =
b 0
δω 0
b 0
0
I
0
0 −T f × T h
0
0
Tbn
0
0
hω 2
n
b
0
0
n
Tb hy1
+ 0
hω1
0
b
n
b y2
0
0
0 δ p n
h y1 δ v n
n
0 δ
b
hω1 δωib
0 b
0
0
w
0 y
w
0 b
I N
(13)
where N is the number of accelerometers in the configuration. The
GF error-state model now includes the nine original error states (as
in the classical INS) plus three additional angular velocity error
states and N additional bias states (one for each accelerometer).
This error model is applicable for any GF configuration. To conclude, and to simplify the notation, we partition the GF error-state
model (13) into block matrices in the following way:
A B
δ x + GW
0 D
δ x =
(14)
F
where
GF-INS ERROR MODEL
For a stationary vehicle and for short periods, a simplified INS
model is given by [10]
0
δ p n 0 0
δ p n 0 I
b
n n
n
δ f
n
b
=
−
×
+
0
0
0
v
T
f
v
T
δ
δ
b
b
b
δ n 0 T n δω
δ n
b
0
0 0
(10)
where δpn is the position error vector, δvn is the velocity error vector, and δϵn is the attitude error vector.
AUGUST 2017
0
0 I
n
b
A = 0 0 −Tb f ×
0
0 0
w
W = y
wb
IEEE A&E SYSTEMS MAGAZINE
0
0
n
B = Tb hy 2 hy1
Tbn
0
0
0
n
Tb hy1 0
G= 0
0
0
h
ω1
0
I N
(15)
h
h
D = ω 2 ω1
0
0
41
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