Global Positioning System Reference
In-Depth Information
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The windup correction is (Wu et al., 1993, p. 95)
d
cos
−
1
d
·
sign
k
·
d
×
d
δ
=
ϕ
(7.3)
d
d
At
a given instant in time, the windup correction
ϕ
cannot be separated from the
un
differenced ambiguities, nor is it absorbed by the receiver clock error because it
is
a function of the receiver and the satellite. In practical applications it is therefore
su
fficient to interpret
x
and
y
as unit vectors along northing and easting and
x
and
y
as
unit vectors in the satellite body coordinate system. Any additional windup error
re
sulting from this redefinition of the coordinate system will also be absorbed by
th
e undifferenced ambiguities. Taken over time, however, the values of
δ
δ
ϕ
reflect the
ch
ange in orientation of receiver and satellite antennas.
The value of the windup correction for single and double differences has an in-
ter
esting connection to spherical trigonometry. Consider a spherical triangle whose
ve
rtices are given by the latitudes and longitudes of the receivers
k
and
m
, and the
sa
tellite. In addition, we assume that GPS transmitting antennas are pointing toward
th
e center of the earth and that the ground receiver antennas are pointing upward. This
as
sumption is usually met in the real world. It can be shown that single difference
wi
ndup correction
[23
Lin
—
3.2
——
No
PgE
ϕ
m
is equal to the spherical excess if the satellite
ap
pears on the left as viewed from station
k
to station
m
, and it equals the negative
sp
herical excess if the satellite appears to the right. The double-differencing windup
co
rrection
ϕ
km
= δ
ϕ
k
δ
− δ
ϕ
pq
km
equals the spherical excess of the respective quadrilateral. The sign
of
the correction depends on orientation of the satellite with respect to the baseline.
Fo
r details, refer to Wu et al. (1993).
The windup correction can be neglected for short baselines because the spher-
ica
l excess of the respective triangles is small. Neglecting the windup correction
m
ight cause problems when fixing the double-difference ambiguities, in particular
fo
r longer lines. The float ambiguities absorb the constant part of the windup correc-
tio
n. The variation of the windup correction over time might not be negligible in float
so
lutions of long baselines.
There is no windup-type correction for the pseudoranges. Consider the simple case
of
a rotating antenna that is at a constant distance from the transmitting source and
th
e antenna plane perpendicular to the direction of the transmitting source. Although
th
e measured phase would change due to the rotation of the antenna the pseudorange
wi
ll not change because the distance is constant.
δ
[23
7.
2.2 Satellite Antenna Phase Center Offset
The satellite antenna phase center offsets are usually given in the satellite-fixed co-
ordinate system
(
x
)
that is also used to express solar radiation pressure (see Section
3.1.4.3). The origin of this coordinate system is at the satellite's center of mass. If
e
de-
notes the unit vector pointing to the sun, expressed in the ECEF coordinate system (x),
then the axes of
(
x
)
are defined by the unit vector
k
(pointing from the satellite toward
the earth's center), the vector
j
=
(
k
×
e
)/
|
k
×
e
|
(pointing along the solar panel axis),