Global Positioning System Reference
In-Depth Information
Table 3.5
Acceleration Forces Perturbing Satellite Orbit
Perturbing Acceleration RMS Orbit Differences over 3 Days (m)
RMS Orbit Determination (m)
Along
Track
Cross
Track
Along
Track
Cross
Track
Radial
Radial
Total
Total
Earth oblateness (C
20
)
1,341
36,788
18,120
41,030
1,147
1,421
6,841
7,054
Moon gravitation
231
3,540
1,079
3,708
87
126
480
504
Sun gravitation
83
1,755
431
1,809
30
13
6
33
C
22
, S
22
80
498
10
504
3
3
4
5
C
nm
, S
nm
(n,m
3..8)
11
204
10
204
4
13
5
15
C
nm
, S
nm
(n,m
4..8)
2
41
1
41
1
2
1
2
C
nm
, S
nm
(n,m
5..8)
1
8
0
8
0
0
0
0
Solar radiation pressure
90
258
4
273
0
0
0
0.001
pressure states consist of a scaling parameter to the a priori solar pressure model and
a Y-body axis acceleration. The Kalman filter clock states include a bias, drift, and
draft rate (for Rubidium only). To avoid numerical instability, the CS Kalman filter
is formulated in
U
-
D
factored form, where the state covariance (e.g.,
P
) is main-
tained as:
T
P UDU
=⋅
⋅
(3.7)
with
U
and
D
being upper triangular and diagonal matrices, respectively [19]. The
U
-
D
filter improves the numerical dynamic range of the CS filter estimates, whose
time constants vary from several hours to several weeks. The CS Kalman time
update has the form:
()
()
()
T
UDU BU
Q
t
~
~
~
[
]
B
U
t
t
$
()()()
T
()()
k
k
(3.8)
t
t
t
=
t
t
$
()
$
k
+
1
k
+
1
k
+
1
k
k
T
D
t
k
k
~
(),
~
()
$
(),
$
()
where
UD
⋅
⋅
and
UD
⋅
⋅
denote the a priori and a posteriori covariance factors,
respectively;
Q
()
denotes the matrix
that maps the process noise to the appropriate state domain. The CS process noises
include the satellite and ground station clocks, troposphere-wet height, solar pres-
sure, and ephemeris velocity (with the latter being in radial, along-track, and
cross-track coordinates [20]). Periodically, the 2SOPS retunes the satellite and
ground station clock process noises, using on-orbit GPS Allan and Hadamard clock
characterization, as provided by the Naval Research Laboratory [21, 22]. The CS
Kalman filter performs scalar measurement updates, with a statistically consistent
test to detect outliers (based on the measurement residuals or innovation process
[18]). The CS measurement model includes a clock polynomial model (up to second
order), the Hopfield/Black troposphere model [23, 24], the IERS station tide dis-
placement model (vertical component only), and periodic relativity and satellite
phase center corrections.
Since a pseudorange measurement is simply the signal transit time between the
transmitting satellite and the receiving monitor station, the CS Kalman filter can
⋅
denotes the state process noise matrix; and
B
()
⋅
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