Global Positioning System Reference
In-Depth Information
[ ]
uL
σ is the predicted standard deviation of the local errors (troposphere, noise, multipath)
on the i
th
signal. The values of the standard deviation of local errors can be read in the UERE
table (Galileo satellites). Typical values are those reported in the following Table (F. Luongo
et al., 2004). The computation is performed using the following interpolation function:
−⋅
=+⋅
10
EL
σ
uL
i
,
()
abe
(5)
i
Where A is a matrix that depends on the elevation angles, σ is the standard deviation
value of the UERE for the i
th
satellite, while a and b can be computed by the following
equation:
T
T
−
1
T
ab
=
(
A
⋅
A
)
⋅
A
σ
⋅
(6)
i
ID
01
02
03
04
05
06
07
08
Elevation [rad]
0.1745
0.2618
0.3491
0.5236
0.6981
0.8727
1.0472 1.5708
σ
i
[m]
1.0300
0.7800
0.6700
0.6000
0.5800
0.5700
0.5600 0.5500
Table 2. UERE table
The predicted standard deviation of total pseudorange error (
uR
σ ) is considered an
internal variable of the HMI Probability Computation Algorithm (HPCA).
The “Position Solution Matrix” module computes the position solution matrix. The main
formula is reported:
(
)
T T
−
KGWGGW
=
(7)
where K is the Position Solution Matrix, G is the Observation Matrix and W is the Weighting
Matrix obtained by inverting the Covariance Matrix.
The “Fault Free Position Error” module computes the characteristic of the position error in
fault free mode (fault free geometry: all the useable satellites). In particular, it evaluates the
standard deviation of the distribution that overbounds the vertical position error and the
variance of the distribution that overbounds the horizontal position error. The principal
formulas are reported as follows:
2
σσ
=
(8)
V
ZZ
2
2
2
2
2
⎛
⎞
σσ σσ
+
−
2
4
XX
YY
XX
YY
ξ
=
+
+
σ
(9)
⎜
⎟
H
⎜
⎟
XY
2
2
⎝
⎠
where σ is the standard deviation of the model (zero-mean normal CDF) used to
overbound the vertical position error in fault free mode.
2
ξ is the variance of the model used to overbound the horizontal position error (along the
semi-mayor axis of the error ellipse) in fault free mode.
The
m
σ components are obtained using this general expression:
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