Global Positioning System Reference
In-Depth Information
In this work the errors in pitch
roll are not modelled inside the KF. This is because
they don't suffer from error growth due to lack of integration operations.
δρ
and
δθ
Therefore there
are no dynamic error states for pitch and roll errors.
Instead, expressions for
δρ
and
δθ
in
˙
V
l
composed of other error states and pitch and roll equations in RISS mechanization are
derived. The equation for velocity error is then re-arranged to accommodate the error terms
belonging to
δ
δρ
and
δθ
.
δρ
The following is a derivation for the expression for
in Equation 21 using Equation 6. Take
the derivative of the error component of
ρ
to give
δρ
:
d
d
f
y
ρ
d
d
f
y
sin
−
1
f
y
1
δρ
=
δ
f
y
=
δ
f
y
=
g
1
f
g
2
δ
f
y
(22)
g
−
A similar operation is performed for the error in roll, keeping in mind that the aim is to find
an alternate expression for
δθ
that contains error terms other than
δθ
and
δρ
. Using Equation
7, the partial derivatives of each component of
θ
in Equation 21 are used to give:
∂
∂
θ
θ
∂
∂
θ
sin
−
1
f
x
+
V
f
ω
z
g
cos
δθ
=
δ
θ
=
−
δ
θ
(23)
ρ
f
x
T
. Taking partial derivatives results in:
δ
θ
=
Where
δ
V
f
δρ δω
z
δ
sin
f
x
+
f
x
V
f
ω
z
g
V
f
g
cos
1
ω
z
g
cos
ρ
1
g
cos
δθ
=
−
1
ρ
δ
V
f
+
δρ
+
ρ
δω
z
+
ρ
δ
f
x
+
V
f
ω
z
g
cos
2
cos
2
ρ
−
ρ
(24)
δρ
Express
and its associated terms in Equation 24 using the components from Equation 22 as
follows:
⎛
⎝
⎞
⎠
sin
sin
+
+
f
x
V
f
ω
z
g
f
x
V
f
ω
z
g
ρ
ρ
1
δρ
=
g
1
f
g
2
δ
f
y
cos
2
ρ
cos
2
ρ
−
⎛
⎝
⎞
⎠
δ
f
x
V
f
ω
z
sin
+
ρ
=
1
f
y
(25)
f
g
2
g
2
cos
2
ρ
−
Express
δ
V
f
contained in
δθ
from Equation 24 in terms of the three velocities along the east,
north and up channels:
V
f
V
u
V
e
∂
∂
V
f
∂
∂
V
f
δ
δ
V
n
δ
=
=
+
+
V
f
V
f
V
f
⎡
⎣
⎤
⎦
=
⎡
⎣
⎤
⎦
δ
V
e
δ
V
e
2
V
u
V
e
V
n
V
u
1
1
V
f
δ
V
n
δ
V
n
=
2
V
e
+
2
V
e
2
V
n
(26)
V
n
+
V
u
δ
δ
V
u
V
u
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