Global Positioning System Reference
In-Depth Information
The following sections contain derivations of the equations for each error state in the model.
These equations use first order terms from the Taylor series expansion of the mechanization
equations.
2.5.2 Position errors
From (Noureldin et al., 2009) the position components of the mechanization equations are
linearized, yielding three error equations for latitude, longitude and altitude.
Neglecting
higher-order terms of the Taylor Series and writing in matrix form gives:
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
=
V
n
⎡
⎣
⎤
⎦
0
0
−
˙
δ
φ
δφ
2
(
M
+
h
)
˙
V
e
tan
φ
δ
λ
V
e
δλ
˙
−
r
l
0
δ
=
(
N
+
h
)
cos
φ
2
cos
(
N
+
h
)
φ
h
δ
h
δ
0
0
0
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
1
M
0
0
δ
V
e
+
h
1
00
δ
V
n
+
(20)
(
N
+
h
)
cos
φ
δ
V
u
0
0
1
Where
M
and
N
are the respective Meridian and normal radii of the curvature of the Earth.
2.5.3 Velocity errors
The velocity components from the mechanization equations are linearized to provide velocity
error equations; these equations are presented in (Noureldin et al., 2009). The velocity errors
are function of errors in position, velocity, and attitude as well as accelerometer stochastic drift
errors.
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
=
V
n
V
e
⎡
⎣
⎤
⎦
φ
+
φ
+
2
ω
e
V
u
sin
2
ω
e
V
n
cos
00
V
e
δ
δφ
(
N
+
h
)
cos
2
φ
V
e
V
n
˙
δ
−
φ −
δλ
V
l
2
ω
e
V
e
cos
00
δ
=
(
N
+
h
)
cos
2
φ
V
u
δ
δ
h
2
g
M
−
2
ω
e
V
e
sin
φ
0
+
h
2
h
⎡
⎣
⎤
⎦
V
n
tan
φ
V
e
tan
φ
V
u
N
V
e
N
⎡
⎣
⎤
⎦
−
h
+
2
ω
e
sin
φ
+
−
ω
e
cos
φ
+
+
N
+
h
N
+
h
+
δ
V
e
2
V
e
tan
φ
V
u
M
V
n
M
−
ω
e
sin
φ
+
−
−
δ
V
n
+
N
+
h
+
h
+
h
2
h
δ
V
u
V
e
N
V
n
M
ω
e
cos
φ
+
2
0
+
+
h
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
+
R
⎡
⎣
⎤
⎦
−
0
f
u
f
n
δρ
δ
f
x
−
f
u
0
f
e
δθ
δ
f
y
l
b
+
(21)
−
f
n
f
e
0
δψ
δ
f
z
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